aa r X i v : . [ m a t h . A T ] D ec G -Global Homotopy Theory andAlgebraic K -Theory Tobias Lenz
Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universit¨atBonn, Endenicher Allee 60, 53115 Bonn, Germany & Max-Planck-Institutf¨ur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
Email address : [email protected] Mathematics Subject Classification.
Primary 55P91, 19D23, Secondary55P48, 18G55
Abstract.
We develop the foundations of G -global homotopy theory as a syn-thesis of classical equivariant homotopy theory on the one hand and globalhomotopy theory in the sense of Schwede on the other hand. Using this frame-work, we are able to construct the G -global algebraic K -theory of symmetricmonoidal categories with G -action, unifying equivariant algebraic K -theory, asoriginally introduced by Shimakawa, and Schwede’s global algebraic K -theory.As an application of the theory, we prove that the G -global algebraic K -theory functor exhibits the symmetric monoidal categories with G -actionas a model of connective G -global stable homotopy theory, generalizing andstrengthening a classical theorem of Thomason. This in particular allows usto deduce the corresponding statements for global and equivariant algebraic K -theory. ontents Introduction 5Chapter 1. Unstable G -global homotopy theory 91.1. Equivariant model structures and Elmendorf’s Theorem 91.2. G -global homotopy theory via monoid actions 311.3. G -global homotopy theory via diagram spaces 621.4. Tameness 871.5. G -global homotopy theory vs. global homotopy theory 104Chapter 2. Coherent commutativity 1132.1. Ultra-commutativity 1132.2. G -global Γ-spaces 1312.3. Comparison of the approaches 158Chapter 3. Stable G -global homotopy theory 1693.1. G -global homotopy theory of G -spectra 1693.2. Connections to unstable G -global homotopy theory 1893.3. G -global spectra vs. G -equivariant spectra 1953.4. Delooping and group completion 197Chapter 4. G -global algebraic K -theory 2094.1. Definition and basic properties 2094.2. G -global algebraic K -theory as a quasi-localization 218Appendix A. Abstract homotopy theory 221A.1. Quasi-localizations 221A.2. Some homotopical algebra 229Bibliography 235 ntroduction Equivariant homotopy theory is concerned with spaces carrying additional ‘sym-metries,’ encoded in the action of a suitable fixed group, and their (co)homologytheories. One of the early successes of the equivariant point of view was the proofof the
Atiyah-Segal Completion Theorem [ AS69 ] using equivariant topological K -theory, generalizing and greatly simplifying Atiyah’s original argument [ Ati61 ]based on non-equivariant K -theory. Subsequently, this motivated the Segal Conjec-ture [ Car92 ] on the stable cohomotopy of classifying spaces, from which much of theoriginal impetus for the development of equivariant stable homotopy theory derived,culminating in Carlsson’s proof [
Car84 ]. Since then, equivariant homotopy theoryhas seen striking applications and connections to other areas of mathematics, rang-ing from the
Sullivan Conjecture [ Sul05 , Chapter 5] (proven by Carlsson [
Car91 ]and in an important special case by Miller [
Mil84 ]) on the relation between fixedpoints and homotopy fixed points, which was motivated by questions about the ho-motopy types of real algebraic varieties, to the celebrated solution of the Kervaireinvariant one problem by Hill, Hopkins, and Ravenel [
HHR16 ] in all dimensionsapart from 126.
Global homotopy theory , as investigated among others by Schwede [
Sch18 ],provides a rigorous framework to talk about ‘uniform equivariant phenomena,’ andin particular provides a natural home for many equivariant (co)homology theoriesthat exist in a compatible way for all suitable groups, like equivariant topological K -theory and equivariant stable bordism. The global formalism has in severalcases led to clean and conceptual descriptions of such uniform phenomena wheredirect descriptions for each individual group are much more opaque, for examplefor equivariant formal group laws [ Hau19a ] or for the zeroth equivariant homotopygroups of symmetric products [
Sch17 ].However, not all G -equivariant cohomology theories come from global ones;in particular, while there is a forgetful functor from the global stable homotopycategory to the G -equivariant one admitting both adjoints, this is not a (Bousfield)localization unless G is trivial, i.e. global homotopy theory is not in any straight-forward way a ‘generalization’ of G -equivariant homotopy theory.The present monograph studies G -global homotopy theory as a synthesis of theabove two approaches; in particular, 1-global homotopy theory recovers Schwede’sglobal homotopy theory, while there exists for every finite group G a forgetful func-tor from G -global to G -equivariant homotopy theory admitting both a fully faithfulleft and a fully faithful right adjoint, yielding right and left Bousfield localizations,respectively. In fact, we develop our theory more generally for all discrete groups G , which for infinite G refines proper equivariant homotopy theory , as developedstably in [ DHL + ]. The idea of G -global homotopy theory has been around forsome time—for example, a specific pointset model of Z / appeared in preliminary versions of [ Sch18 ], while Σ n -global weak equivalencesfor n ≥ Bar20 ] for the study of operads in unsta-ble global homotopy theory—, but this seems to be the first time the theory isdeveloped systematically.While there should also be a notion of G -global homotopy theory for (compact)Lie groups, we restrict our attention to discrete groups here. This in particularallows us to construct models of a more combinatorial nature, which is crucial forour main application: Equivariant algebraic K -theory. Our study of G -global homotopy theory ismotivated by algebraic K -theory, and more precisely by two refinements of the alge-braic K -theory of small symmetric monoidal categories as originally introduced byShimada and Shimakawa [ SS79 ] using Segal’s theory of (special) Γ-spaces [
Seg74 ]:On the one hand, Shimakawa [
Shi89 ] generalized Segal’s theory to constructthe G -equivariant algebraic K -theory of small symmetric monoidal categories with G -action; his construction has been revisited in recent years among others by Mer-ling [ Mer17 ] and her coauthors [
MMO17 ].On the other hand, Schwede [
Sch19b ] recently introduced the global algebraic K -theory of so-called parsummable categories ; he describes how small symmetricmonoidal categories yield parsummable categories, and we have shown in [ Len20a ]that this accounts for all examples up to homotopy.These two approaches generalize the original construction into two differentdirections, and in particular neither is a special case of the other—for example,Schwede’s theory takes less general inputs, but yields more structure on the output.Accordingly, any direct comparison of the two constructions would be rather weak,only capturing a fraction of each of them.Here we take a different route to clarifying the relation between the two ap-proaches by realizing them as facets of a more general construction: using theframework of G -global homotopy theory developed in this monograph, we intro-duce G -global algebraic K -theory , which for G = 1 again recovers global algebraic K -theory, while for general G it refines the equivariant construction. Togetherwith a basic compatibility property of G -global algebraic K -theory under change ofgroups, this in particular shows that Schwede’s global algebraic K -theory of a smallsymmetric monoidal category C forgets to the G -equivariant algebraic K -theory of C equipped with the trivial action, and we suggest to think of the existence of G -global algebraic K -theory as a general comparison of the above two approaches. A G -global Thomason Theorem. A classical result of Thomason says thatall connective stable homotopy types arise as K -theory spectra of small symmet-ric monoidal categories; even stronger, he proved as [ Tho95 , Theorem 5.1] thatthe K -theory functor expresses the homotopy category of connective spectra as alocalization of the 1-category of small symmetric monoidal categories.Building on our results in [ Len20a ], we prove the following G -global general-ization of Thomason’s theorem as the main result of this monograph: Theorem A . For any discrete group G , the G -global algebraic K -theory con-struction exhibits the quasi-category of connective G -global stable homotopy types asa quasi-categorical localization of the category G -SymMonCat of small symmetricmonoidal categories with G -action (through strong symmetric monoidal functors). NTRODUCTION 7
This in particular immediately yields the corresponding statements for globaland equivariant algebraic K -theory: Theorem B . Schwede’s global algebraic K -theory functor exhibits the quasi-category of connective global stable homotopy types as a quasi-categorical localizationof both SymMonCat and of the category
ParSumCat of parsummable categories.
Theorem C . For any finite G , Shimakawa’s equivariant algebraic K -theoryfunctor exhibits the quasi-category of connective G -equivariant stable homotopytypes as a quasi-categorical localization of G -SymMonCat . Theorem B had been conjectured by Schwede in [
Sch19b ]; to the best of ourknowledge, also Theorem C had not been proven before.In fact, our methods yield a bit more: Mandell [
Man10 ] strengthened Thoma-son’s result to an equivalence between symmetric monoidal categories (up to weakhomotopy equivalences) and special Γ-spaces; in view of Segal’s description of thepassage from special Γ-spaces to connective spectra, Mandell called this a ‘non-group-completed’ version of Thomason’s theorem. Conversely, his result then givesa conceptual description of algebraic K -theory as a ‘higher group completion.’Generalizing his result, we also prove ‘non-group-completed’ versions of Theo-rems A–C, in particular showing that small symmetric monoidal categories modelall ultra-commutative monoids in the sense of [ Sch18 ], and that small symmet-ric monoidal categories with G -action are equivalent to Shimakawa’s special Γ- G -spaces. This way, we also get conceptual descriptions of equivariant, global, and G -global algebraic K -theory as ‘higher equivariant (global; G -global) group com-pletions,’ which we can in particular view as evidence that these are the ‘right’generalizations of classical algebraic K -theory.While the study of global versions of Γ-spaces, which is a key ingredient to ourproof of Theorem B, naturally leads to G -global homotopy theory, it is interestingthat ( G -)global techniques are also central to our proof of the purely equviariantTheorem C. In particular, the argument given here makes crucial use of modelsbased on Schwede’s notion of ‘ultra-commutativity,’ and I am not aware of a simpleway to bypass them to obtain a standalone proof of Theorem C. Outline.
Chapter 1 introduces and compares various models of unstable G -global homotopy theory , laying the foundations for the results established in laterchapters. We also compare our approach to usual proper equivariant homotopy the-ory (Theorem 1.2.83) and to Schwede’s model of unstable global homotopy theoryin terms of orthogonal spaces (Theorem 1.5.23).We then study various notions of ‘ G -globally coherently commutative monoids’in Chapter 2, in particular G -global versions of Γ-spaces or ultra-commutativemonoids. As the main result of this chapter (Theorem 2.3.1) we prove that thesemodels are equivalent, harmonizing Schwede’s global approach with the classicalequivariant theory.Chapter 3 is concerned with stable G -global homotopy theory . We introduce gen-eralizations of Hausmann’s global model structure [ Hau19b ] to symmetric spectrawith G -action, whose weak equivalences for finite G refine the usual G -equivariantstable weak equivalences [ Hau17 ]. Using this, we then prove a G -global strength-ening of a classical result due to Segal (Theorem 3.4.22): any G -global Γ-space canbe delooped to a connective G -global spectrum, and this provides an equivalence INTRODUCTION between so-called very special G -global Γ-spaces on the one hand and connective G -global spectra on the other hand.In Chapter 4 we introduce G -global algebraic K -theory and compare it to bothequivariant and global algebraic K -theory. Finally, we use almost all of the theorydeveloped in the previous chapters together with our results in [ Len20a ] to proveTheorems A–C.
Acknowledgements.
The results in this monograph were obtained as part ofmy PhD thesis at the University of Bonn. I would like to thank my advisor StefanSchwede for suggesting global and equivariant algebraic K -theory as a thesis topic,for various instructive discussions about the results of this paper, and for helpfulremarks on previous versions of parts of the material presented here.This work began as a project to prove a global version of Thomason’s theorem.I am indebted to Markus Hausmann, whose earlier suggestion that there shouldbe a notion of G -global homotopy theory proved to be an illuminating perspectivecompletely changing the form of this monograph and leading to the proof of theequivariant version of Thomason’s theorem.Finally, I would like to thank the Max Planck Institute for Mathematics inBonn for their hospitality and generous support.The author is an associate member of the Hausdorff Center for Mathematics,funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Founda-tion) under Germany’s Excellence Strategy (GZ 2047/1, project ID390685813).HAPTER 1 Unstable G -global homotopy theory In this chapter we will introduce several models of unstable G -global homotopytheory , generalizing Schwede’s unstable global homotopy theory [ Sch18 , Chapter 1].These models are already geared towards the study of G -global algebraic K -theory,and in particular, while we will be ultimately interested in stable G -global homotopytheory and in the theory of G -global infinite loop spaces, the comparisons provenhere will be instrumental in establishing results on the latter. Let G be a discrete group. In unstable G -equivariant homotopy theory one isinterested in G -spaces or G -simplicial sets up to so-called (genuine) G -weak equiv-alences , i.e. those G -equivariant maps that induce weak homotopy equivalence on H -fixed points for all subgroups H ⊂ G . More generally, one can consider the F -weak equivalences for any collection F of subgroups of G , i.e. those maps thatinduce weak homotopy equivalence on H -fixed points for H ∈ F . If F = A ℓℓ is thecollection of all subgroups, this recovers the previous notion; at the other extreme,if F consists only of the trivial group, then the F -weak equivalences are preciselythe underlying weak homotopy equivalences. We will at several points be interestedin proper equivariant homotopy theory , where one considers the class F in of finite subgroups of G ; of course, F in = A ℓℓ if G is finite, but for infinite G these differ.It is a classical result that there exists for any G and F as above a modelstructure on the category G -SSet of G -simplicial sets whose weak equivalences andfibrations are precisely those maps f such that f H is a weak homotopy equivalenceor Kan fibration, respectively, for all H ∈ F , and similarly for topological spaces.A particularly conceptual approach to the construction of these model structuresis due to Stephan [ Ste16 ], who gave a list of mild axioms for a general cofibrantlygenerated model category C to support an analogous model structure on G - C .Our first approach to unstable G -global homotopy theory will rely on a gener-alization of this to actions of simplicial monoids : Proposition . Let M be a simplicial monoid and let F be a family offinite subgroups of M . Then the category M -SSet of M -objects in SSet admitsa (unique) model structure such that a map f : X → Y is a weak equivalence orfibration if and only if f G : X G → Y G is a weak homotopy equivalence or Kanfibration, respectively, for all G ∈ F .We call this the F -model structure (or, if F is clear from the context, the M -equivariant model structure , or even simply the equivariant model structure ) andits weak equivalences the F -weak equivalences (or simply M -weak equivalences ; equivariant weak equivalences ). It is simplicial, combinatorial, and proper. A
90 1. UNSTABLE G -GLOBAL HOMOTOPY THEORY possible set of generating cofibrations is given by I = { M/G × ∂ ∆ n ֒ → M/G × ∆ n : n ≥ , G ∈ F} and a possible set of generating acyclic cofibrations is given by J = { M/G × Λ nk ֒ → M/G × ∆ n : 0 ≤ k ≤ n, G ∈ F} . Moreover, filtered colimits are homotopical in M -SSet . The condition on the finiteness of the subgroups G is not necessary for theexistence of the model structure, but it will simplify several arguments. It is howevercrucial that we only test weak equivalences and fibrations with respect to discretesub groups .The proof of Proposition 1.1.1 will be given in Subsection 1.1.3 as an instanceof a more general theorem. However, we already note: Lemma . The weak equivalences of the above model structure are closedunder finite products and small coproducts.
Proof.
As fixed points commute with products and coproducts, this is imme-diate from the corresponding statement for ordinary simplicial sets. (cid:3)
As in the classical case, we willconstruct our desired model structure on M -SSet as a transferred model structure : Definition . Let D be a complete and cocomplete category, let C be amodel category and let F : C ⇄ D : U be an (ordinary) adjunction. The modelstructure transferred along F ⊣ U on D is the (unique if it exists) model structurewhere a morphism f is a weak equivalence or fibration if and only if U f is.
Lemma . Let F : C ⇄ D : U be as above, and assume the transferredmodel structure on D exists. (1) If C is right proper, then so is D . (2) Assume that C is a simplicial model category, D is enriched, tensored,and cotensored over SSet , and that F ⊣ U can be enriched to a simplicialadjunction. Then D is a simplicial model category. Proof.
We show the second statement, the proof of the first one being similarbut easier. For this we only have to verify Quillen’s axiom (SM7), i.e. that for anycofibration i : K ֒ → L in SSet and any fibration p : X → Y in D the ‘pullbackcorner map’ ( i ∗ , p ∗ ) : X L → X K × Y K Y L is a fibration and that this is acyclic if at least one of i and p is. By definition,this can be detected after applying U , and as U is a simplicial right adjoint theresulting map agrees up to conjugation by isomorphisms with( i ∗ , U ( p ) ∗ ) : U ( X ) L → U ( X ) K × U ( Y ) K U ( Y ) L . As U is right Quillen, U ( p ) is a fibration, acyclic if p is. Hence the claim followsfrom the fact that C is simplicial. (cid:3) Crans and Kan gave useful criteria for the transferred model structure to exist.In order to state them, we recall that for a class I of morphisms in a cocompletecategory C a relative I -cell complex is a map that can be written as a transfinitecomposition of pushouts of maps in I . Moreover, an object is an I -cell complex if the .1. EQUIVARIANT MODEL STRUCTURES AND ELMENDORF’S THEOREM 11 unique map from the initial object is a relative I -cell complex, and it is I -cofibrant if it is a retract of some I -cell complex; note that if C is a cofibrantly generatedmodel category and I its set of generating cofibrations, then the I -cofibrant objectsare precisely the cofibrant objects by Quillen’s Retract Argument.Finally we recall that a set I of morphisms is said to permit the small objectargument if the sources of maps in I are small with respect to I -cell complexes. Proposition . Let D be a complete and cocomplete category, let C be acofibrantly generated model category, I a set of generating cofibrations, J a set ofgenerating acyclic cofibrations, and let F : C ⇄ D : U be an (ordinary) adjunction. Assume the following: (1) The sets
F I and
F J permit the small object argument. (2)
Any map obtained as transfinite composition of pushouts of maps of theform F ( j ) with j ∈ J a generating acyclic cofibration, is a weak equiva-lence (i.e. sent by U to a weak equivalence).Then the transferred model structure exists. Moreover, it is cofibrantly generatedwith sets of generating cofibrations F I and generating acylic cofibrations
F J . Proof.
Crans [
Cra95 , Theorem 3.3] proved a slightly weaker statement; theabove appears as [
Hir03 , Proposition 10.2.14], where this is attributed to Kan. (cid:3)
Remark . Assume that U preserves λ -filtered colimits for some regu-lar cardinal λ and assume that all generating cofibrations respectively generat-ing acyclic cofibrations are maps between small objects (e.g. when C is locallypresentable). Then it is a standard fact—see e.g. [ Cra95 , discussion after Defini-tion 3.2]—that Condition (1) of the above proposition holds. Indeed, we can choosea κ > λ such that all sources and targets of maps in I and J are κ -small and aneasy adjointness argument shows that F preserves κ -smallness.We will now give another criterion to transfer model structures that applies inthe situations relevant in this monograph, and moreover allows us to prove thatthe induced Quillen adjunction is even a Quillen equivalence . This will need somepreparations.1.1.1.1. Cell induction.
In the proof of our transfer criterion as well as at severalpoints later, we will use the strategy of ‘cell induction.’ In order to avoid repeatingthe same standard argument over and over, we formalize it once and for all.
Lemma . Let C be a cocomplete category, let I be a class of morphismsand let S be a class of objects in C such that the following conditions are satisfied: (1) ∅ ∈ S (2) If (1.1.1) A BC D i is a pushout with C ∈ S and i ∈ I , then D ∈ S . (3) S is closed under filtered colimits.Then S contains all I -cell complexes. If in addition also G -GLOBAL HOMOTOPY THEORY (4) S is closed under retracts,then S contains all I -cofibrant objects. In particular, if C is a cofibrantly generatedmodel category and I a set of generating cofibrations, then S contains all cofibrantobjects. Proof.
It is enough to prove the first statement. For this let X be an I -cellcomplex, i.e. we have an ordinal α and a functor X • : { β < α } → C such that thefollowing conditions hold:(A) X = ∅ (B) For each β such that β + 1 < α , the map X β → X β +1 is a pushout ofsome generating cofibration i ∈ I .(C) If β < α is a limit ordinal, then the maps X γ → X β for γ < β exhibit X β as a (filtered) colimit.If α is a limit ordinal, we extend this to { β ≤ α } via X α := colim β<α X β togetherwith the obvious structure maps; if α is a successor ordinal instead, we replace α by its predecessor. In both cases we have a functor X • : { β ≤ α } → C satisfyingconditions (A)–(C) above (with ‘ < ’ replaced by ‘ ≤ ’) and such that X α = X . Wewill prove by transfinite induction that X β ∈ S for all β ≤ α which will then implythe claim.By Conditions (1) and (A) we have that X = ∅ ∈ S . Now assume β > γ < β . If β is a successor ordinal, β = γ + 1,then Conditions (2) and (B) together with the induction hypothesis imply that X β ∈ S . On the other hand, if β is a limit ordinal, then the maps X γ → X β for γ < β express X β as a filtered colimit of elements of S by Condition (C) togetherwith the induction hypothesis. Thus, Condition (3) immediately implies the claim,finishing the proof. (cid:3) The following corollary in particular formalizes the usual proof of the classicalElmendorf Theorem for finite groups:
Corollary . Let C be a cocomplete category and let D be a left propermodel category such that filtered colimits in it are homotopical. Let I be any collec-tion of morphisms in C , and let F, G : C → D be functors together with a naturaltransformation τ : F ⇒ G . Assume the following: (1) τ ∅ is a weak equivalence and for every map ( X → Y ) ∈ I both τ X and τ Y are weak equivalences. (2) If A BC D i is a pushout in C along a map i ∈ I , then its images under F and G arehomotopy pushout squares. (3) F and G preserve filtered colimits up to weak equivalence, i.e. for any fil-tered category J and any diagram X : J → C the canonical maps colim J F ◦ X → F (colim J X ) and colim J G ◦ X → G (colim J X ) are weak equiva-lences. .1. EQUIVARIANT MODEL STRUCTURES AND ELMENDORF’S THEOREM 13 Then τ X is a weak equivalence for every I -cofibrant X ∈ C . In particular, if C is acofibrantly generated model category and I is a set of generating cofibrations, then τ X is a weak equivalence for all cofibrant X . Proof.
Let S be the class of objects X ∈ C such that τ X is a weak equiv-alence. It will suffice to verify the conditions of the previous lemma. Indeed,Condition (1) of the lemma is in immediate consequence of our Condition (1).In order to verify Condition (2) of the lemma, we consider a pushout as in(1 . . GA GBF A F BGC GDF C F D
GiF i with all the diagonal maps coming from τ . The assumption C ∈ S implies that thefront-to-back map at the lower left corner is a weak equivalence, and our Condition(1) tells us that the two top front-to-back maps are weak equivalences. Moreover,both front and back square are homotopy pushouts by Condition (2). We concludethat also the lower right front-to-back map is a weak equivalence, just as desired.To verify Condition (3) of the lemma, we observe that for any filtered category J and any X • : J → C the map τ colim j ∈ J X j fits into a commutative diagramcolim j ∈ J F ( X j ) colim j ∈ J G ( X j ) F (colim j ∈ J X j ) G (cid:0) colim j ∈ J X j (cid:1) colim j ∈ J τ Xj τ colim j ∈ J Xj where the vertical maps are the canonical comparison maps and hence weak equiv-alences by our Condition (3). If now all X i lie in S , then the top horizontal map isa filtered colimit of weak equivalences and hence a weak equivalence by assumption,proving Condition (3) of the lemma.Finally, Condition (4) for S is automatic as weak equivalences in any modelcategory are closed under retracts. This finishes the proof. (cid:3) Remark . In most cases of interest to us, C = D and either F or G willbe the identity. In this situation, the corresponding half of Condition (3) holds fortrivial reasons and so does the respective half of (2) by left properness. Remark . If C is also a model category, F and G are left Quillen and I consists of cofibrations, then the second and third condition of the corollary holdautomatically. However, in most cases where we want to apply the corollary atleast one of the functors will not even be cocontinuous.1.1.1.2. Homotopy pushouts.
For our transfer criterion we will need some gen-eral results about closure properties of homotopy pushout squares. We begin withthe 1-categorical situation: G -GLOBAL HOMOTOPY THEORY Lemma . Let C be a cocomplete category, let I be a filtered category witha minimal element , let X • , Y • : I → C be functors and let τ : X • ⇒ Y • be a naturaltransformation such that for each i ∈ I the square X X i Y Y iτ τ i is a pushout. Then the induced square (1.1.2) X colim i ∈ I X i Y colim i ∈ I Y iτ colim i ∈ I τ i is also a pushout. Proof.
We decompose (1 . .
2) as X colim i ∈ I X colim i ∈ I X i Y colim i ∈ I Y colim i ∈ I Y i . τ colim i ∈ I τ colim i ∈ I τ i The right hand square is a pushout as a colimit of pushout squares. But the twoleft hand horizontal maps are isomorphisms as filtered categories have connectednerve, so also the total rectangle is a pushout as desired. (cid:3)
Here is the homotopy theoretic analogue of the previous result:
Lemma . Let C be a left proper model category such that filtered colimitsin C are homotopical, let I be a filtered category with a minimal element , let X • , Y • : I → C be functors and let τ : X • ⇒ Y • be a natural transformation suchthat for each i ∈ I the square (1.1.3) X X i Y Y iτ τ i is a homotopy pushout. Then the induced square (1.1.4) X colim i ∈ I X i Y colim i ∈ I Y iτ colim i ∈ I τ i is also a homotopy pushout. Proof.
We choose a factorization X κ −→ P ϕ −→ Y of τ : X → Y into acofibration followed by a weak equivalence. Next, we choose for each i ∈ I an .1. EQUIVARIANT MODEL STRUCTURES AND ELMENDORF’S THEOREM 15 honest pushout(1.1.5) X X i P P iκ κ i in such a way that for i = 0 the lower horizontal map P → P is the identity.By the universal property of pushouts there is a unique way to extend the chosenmaps P → P i to a functor P • : I → C in such a way that the κ i assemble intoa natural transformation κ : X • ⇒ P • . Moreover, we define ϕ i : P i → Y i to bethe map induced by the square (1 . .
3) via the universal property of the pushout(1 . . ϕ i is a weak equivalence by definition of homotopy pushouts inleft proper model categories. Moreover, the ϕ i obviously assemble into a naturaltransformation ϕ : P • ⇒ Y • such that τ = ϕκ .Thus we can factor (1 . .
4) as X colim i ∈ I X i P colim i ∈ I P i Y colim i ∈ I Y i . κ colim i ∈ I κ i ϕ ∼ colim i ∈ I ϕ i The top square is a pushout by the previous lemma and the left hand vertical mapis a cofibration by construction, so it is a homotopy pushout by left properness of C .On the other hand, the lower left vertical map is a weak equivalence by construction,whereas the lower right vertical map is a weak equivalence as a filtered colimit ofweak equivalences. Thus the total square is a homotopy pushout, finishing theproof. (cid:3) Now we can prove the desired closure properties:
Proposition . Let C be a left proper model category, let D be any co-complete category, and let U : D → C be any functor preserving filtered colimits upto weak equivalence. Assume moreover that filtered colimits in C are homotopical.Let us write H for the class of morphisms i : A → B in D such that U sendsall pushouts along i to homotopy pushouts. Then H is closed under pushouts,transfinite compositions, and retracts in D . Proof.
Consider any diagram
A BC DE F f in D with f ∈ H and such that both squares are pushouts. Applying U to it thenyields by assumption a diagram in C such that both the top square as well as the G -GLOBAL HOMOTOPY THEORY big rectangle are homotopy pushouts. It follows that also the bottom square is ahomotopy pushout, and hence we conclude that H is indeed closed under pushouts.Next, assume that we have a diagram A ′ A A ′ B ′ B B ′ f ′ i f r f ′ j s exhibiting f ′ as a retract of some f ∈ H and consider any pushout(1.1.6) A ′ B ′ C ′ D ′ . α ′ f ′ β ′ We now construct a commutative cube
A BA ′ B ′ C DC ′ D ′ . fi f ′ jk ℓ as follows: starting from A BA ′ B ′ C ′ D ′ fi f ′ j we form the pushout C of C ′ ← A ′ → A , yielding the left hand square. With thisconstructed, we can also form the pushout D of C ← A → B , yielding the backsquare. Finally, the map ℓ : D ′ → D is induced by the functoriality of pushoutsfrom i, j, k .On the other hand we also have a commutative cube A ′ B ′ A BC ′ D ′ .C D f ′ r f st u Here t is induced via the universal property of the pushout C by α ′ r : A → C ′ andid : C ′ → C ′ ; note that this is indeed well-defined as α ′ ri = α ′ id = id α ′ , and itmakes the left hand square commute by construction. Afterwards, u is induced viathe functoriality of pushouts by r, s, t . .1. EQUIVARIANT MODEL STRUCTURES AND ELMENDORF’S THEOREM 17 By assumption we have ri = id and sj = id and by construction also tk = id;the universal property of the pushout D ′ implies that also uℓ = id. Altogether,we have exhibited the pushout (1 . .
6) as retract of a pushout along f ∈ H . Inparticular we see that applying U to (1 . .
6) yields a square that is a retract ofa homotopy pushout, hence itself a homotopy pushout. We conclude that H isindeed closed under retracts.Finally, we have to prove that H is closed under transfinite compositions.Analogously to the argument in the proof of Lemma 1.1.7 this amounts to thefollowing: given any ordinal α and a functor X • : { β ≤ α } → D such that(A) For each β < α , the map X β → X β +1 is in H ,(B) If β ≤ α is a limit ordinal, then the maps X γ → X β for γ < β express X β as (filtered) colimit,then the induced map X → X α is in H again. So let us consider any pushout(1.1.7) X X α C D f g in D ; we have to prove that U sends this to a homotopy pushout.For this we define C = C , τ = f , and then choose for each β ≤ α a pushout(1.1.8) X X β C C βτ τ β in such a way that for β = 0 the lower horizontal map is the identity and for β = α the chosen pushout is given by (1 . . C → C β as to yield a functor C • : { β ≤ α } → C so that the τ β assemble into a natural transformation X • ⇒ C • .We observe that by Lemma 1.1.11 together with the uniqueness of colimits, themaps C γ → C β , γ < β exhibit C β as colimit for every limit ordinal β ≤ α .We will now prove by transfinite induction that the square (1 . .
8) is sent by U to a homotopy pushout for each β ≤ α . Specializing to β = α then precisely provesthe claim.For β = 0 both horizontal maps are identity arrows, hence the same holdsafter applying U . In particular, the resulting square is a homotopy pushout. Nowassume β > γ < β .If β is a successor ordinal, β = γ + 1, then we decompose our square into X X γ X β C C γ C β . τ τ γ τ β Both the left hand square as well as the total rectangle are pushouts by construc-tion, hence so is the right hand square. The left hand square is sent by U to ahomotopy pushout by induction hypothesis; on the other hand ( X γ → X β ) ∈ H by assumption, hence also the right hand square is sent to a homotopy pushout.But then the whole square is sent to a homotopy pushout, just as desired. G -GLOBAL HOMOTOPY THEORY On the other hand, if β is a limit ordinal, then we decompose our square into X colim γ<β X γ X β C colim γ<β C γ C β . f colim γ<β τ γ τ β We have seen above that the two right hand horizontal maps are isomorphisms. To-gether with the assumption that U preserve filtered colimits up to weak equivalence,we conclude that in the diagram U ( X ) colim γ<β U ( X γ ) U ( X β ) U ( C ) colim γ<β U ( C γ ) U ( C β ) . U ( f ) colim γ<β U ( τ γ ) U ( τ β ) the right hand horizontal maps are weak equivalences. Thus it suffices to prove thatthe left hand square is a homotopy pushout, which is just an instance of the previouslemma. We conclude that H is indeed stable under transfinite composition, whichfinishes the proof. (cid:3) Proof of the criterion.
We are now ready for the version of the transferprinciple that will be used throughout this monograph:
Proposition . Let C be a left proper model category cofibrantly generatedby maps between small objects, such that filtered colimits in C are homotopical. Let I be a set of generating cofibrations between small objects and let (1.1.9) F : C ⇄ D : U be an adjunction, where D is a complete and cocomplete category. Assume thefollowing: (1) C admits a set J of generating acyclic cofibrations between small objectsand with cofibrant sources . (2) η ∅ is a weak equivalence, and for each generating cofibration ( X → Y ) ∈ I both η X and η Y are weak equivalences. (3) U sends any pushout square F A F BC D
F i in D , where i ∈ I is a generating cofibration of C , to a homotopy pushoutin C . (4) U preserves filtered colimits up to weak equivalence and there exists someregular cardinal κ such that U preserves κ -filtered colimits (up to isomor-phism).Then the transferred model structure on D exists and it is left proper, cofibrantlygenerated with set of generating cofibrations F I and set of generating acyclic cofi-brations
F J , and filtered colimits in D are homotopical.Moreover, the adjunction (1 . . is a Quillen equivalence. Proof.
We begin with the following crucial observation: .1. EQUIVARIANT MODEL STRUCTURES AND ELMENDORF’S THEOREM 19
Claim. If X ∈ C is cofibrant, then η X : X → U F X is a weak equivalence.
Proof.
It suffices to verify the conditions of Corollary 1.1.8 (applied to the unit η : id ⇒ U F ). Indeed, our Condition (2) is just a reformulation of Condition (1) ofthe corollary. Moreover, Condition (2) is a special case of our Condition (3) as theleft adjoint F preserves all colimits and hence in particular pushouts. Likewise, F inparticular preserves filtered colimits and moreover U preserves filtered colimits upto weak equivalence by our Condition (4), verifying Condition (3) of the corollary.The claim follows. △ To see now that the transferred model structure exists and is cofibrantly gen-erated by
F I and
F J it suffices to verify the conditions of Proposition 1.1.5. ByRemark 1.1.6 and the second half of Condition (4) it is enough that relative
F J -cellcomplexes be weak equivalences in D , i.e. sent under U to weak equivalences in C .The assumptions guarantee that transfinite compositions of weak equivalences areweak equivalences, so it suffices to prove this for pushouts of maps in F J .Let us consider the class H of those maps i ′ : A ′ → B ′ in D such that U sendspushouts along them to homotopy pushouts, i.e. those maps such that the analogueof Condition (3) holds for them. Proposition 1.1.13 then precisely tells us that H is closed under pushouts, transfinite compositions, and retracts.As F preserves pushouts, transfinite compositions, and retracts (being a leftadjoint functor), it follows that also F − ( H ) is closed under all of these. As itcontains all i ∈ I by assumption, it follows by the characterizations of cofibrationsin a cofibrantly generated model category, that F − ( H ) contains all cofibrationsof C ; in particular it contains J . Hence if j ∈ J is a generating acyclic cofibration,and we have any pushout square(1.1.10) F A F BC D,
F jk then applying U to this yields a homotopy pushout in D . But U F j is a weakequivalence, as A (and hence B ) were assumed to be cofibrant, so that η A , η B areweak equivalences by the claim above. It follows that U k is a weak equivalence andhence by definition so is k . Altogether, we have verified the conditions of Propo-sition 1.1.5, proving that the transferred model structure exists and is cofibrantlygenerated by F I and
F J .But with this established, we conclude by the same argument (this time appliedin D ) from the closure properties of H that U sends pushouts along cofibrationsin D to homotopy pushouts. Hence, if we have any pushout A ′ B ′ C D f ∼ i ′ g where i is a cofibration and f is a weak equivalence, then its image under U isa homotopy pushout. As before, we conclude that U g is a weak equivalence andhence so is g , i.e. D is left proper. Similarly, we conclude from Condition (4) andthe corresponding statement for C that filtered colimits in D are homotopical.Altogether, we have proven the first part of the proposition. G -GLOBAL HOMOTOPY THEORY For the second part we now simply observe that U by definition preserves andreflects weak equivalences. Thus it suffices to prove that for any cofibrant X ∈ C the unit map η X : X → U F X is a weak equivalence. But this is precisely thecontent of the above claim, finishing the proof. (cid:3)
Remark . The above proof bears some similarity to the argument usedby Barwick and Kan in the proof of [
BK12 , Theorem 6.1] in that it first establishesthat the unit is a weak equivalence on cofibrant objects and then uses this to showthat the generating acyclic cofibrations are sent to weak equivalences.However, in order to show that in fact all transfinite compositions of pushouts ofgenerating cofibrations are sent to weak equivalences, Barwick and Kan single out acertain explicit class of maps such that pushouts along them are sent to homotopypushouts under the right adjoint U , and then show that this class contains thegenerating cofibrations and satisfies suitable closure properties. Remark . In almost all of the cases in which we want to apply the aboveproposition the following stronger version of Condition (4) will be satisfied:(4 ′ ) U preserves filtered colimits. Remark . In all applications of Proposition 1.1.14 in this paper, thecategory C will be locally presentable (and hence a combinatorial model category).In particular, every object of C is small, so that all the smallness assumptions ofthe above theorem are automatically satisfied. We will freely and without furthermention use this simplification.In fact, also D will be locally presentable. Since any right adjoint functorbetween locally presentable categories is accessible, this means that the secondhalf of Condition (4) becomes vacuous. However, in almost all cases the strongercondition (4 ′ ) will hold while in the remaining case accessibility of U is trivial toestablish. Accordingly, we will not use this further simplification.We also note the following detection criterion for homotopy pushouts: Lemma . In the above situation, U creates homotopy pushouts. Proof.
We have already seen in the above proof that U sends pushouts alongcofibrations to homotopy pushouts. If we are now given any square(1.1.11) A BC D in D , then we factor A → B into a cofibration A → H followed by a weak equiva-lence H → B . We then get a commutative cube A BA HC DC P = ∼ =.1. EQUIVARIANT MODEL STRUCTURES AND ELMENDORF’S THEOREM 21 where the front face is a pushout. All front-to-back maps are weak equivalencesexcept possibly the lower right one, which is (as D is left proper) a weak equivalenceif and only if (1 . .
11) is a homotopy pushout.If we now apply U to this diagram, we get a commutative cube in C whosefront face is a homotopy pushout (by the above observation) and all of whose front-to-back maps except possibly the lower right one are weak equivalences (as U ishomotopical). We conclude that the lower right front-to-back map of this cube isa weak equivalence if and only if also the back square is a homotopy pushout.If now (1 . .
11) is a homotopy pushout, then P → D and hence also U ( P → D )is a weak equivalence. We conclude from the above that U preserves homotopypushouts.On the other hand, if the back square of the cube obtained by applying U ,i.e. the result of applying U to (1 . . U ( P → D ) is also a weak equivalence. As U reflects weak equivalences, so is P → D , and we conclude that U also reflects homotopy pushouts. This finishes theproof. (cid:3) Remark . In fact, any homotopical functor that is part of a Quillenequivalence between arbitrary (i.e. not necessarily left proper) model categoriespreserves and reflects all homotopy colimits and limits, as one for example sees bypassing to the associated derivators or associated quasi-categories.However, this argument requires quite a lot of non-trivial input (most notablythe identification of homotopy colimits in the model categorical sense with thosein the derivator theoretic or quasi-categorical sense), so we have decided to insteadgive a simple direct proof for this simple lemma.
We will now usethe above criterion to prove a very general version of Elmendorf’s Theorem, whichwill in particular allow us to construct the desired model structure on M -SSet .For this let us fix a complete and cocomplete category C that is in additionenriched, tensored, and cotensored over SSet . Our construction is based on thefollowing definition, which is similar to the ‘cellularity conditions’ of [
Ste16 , Propo-sition 2.6]:
Definition . We call a family (Φ i ) i ∈ I of enriched functors C → SSet cellular if the following conditions are satisfied:(1) Each Φ i preserves tensors as well as filtered colimits.(2) Each Φ i is corepresentable in the enriched sense.(3) For each i, j ∈ I , n ≥
0, and some (hence any) X i corepresenting Φ i ,the functor Φ j sends pushouts along ∂ ∆ n ⊗ X i → ∆ n ⊗ X i to homotopypushouts in SSet (with respect to the weak homotopy equivalences).
Example . We will prove in Lemma 1.1.33 that for any simplicial monoid M and any collection F of finite subgroups of M the family (cid:0) (–) G : M -SSet → SSet (cid:1) G ∈F of fixed point functors is cellular. Remark . If one is careful, one can weaken the condition that each Φ j preserve filtered colimits to the assumption that it preserves transfinite composi-tions along pushouts of maps of the form ∂ ∆ n ⊗ X i → ∆ n ⊗ X i , which is closerto [ Ste16 , Proposition 2.6]. For this version, the finiteness of the subgroups is notnecessary in the previous example. G -GLOBAL HOMOTOPY THEORY However, the above definition fits more naturally into our framework, is simplerto formulate, and it has the additional advantage that we get a model category inwhich filtered colimits are homotopical.
Definition . Let (Φ i ) i ∈ I be a cellular family. An orbit category for Φ • is a full simplicial subcategory O Φ • ⊂ C satisfying the following conditions:(1) For each X ∈ O Φ • the functor corepresented by X on C is isomorphic toone of the Φ i .(2) Each Φ i is corepresented on C by some X i ∈ O Φ • . Example . There are two extreme cases of the above definition: on theone hand, we can choose for each Φ i some X i corepresenting it and then take O Φ • to be the full simplicial subcategory spanned by these objects; on the other hand,we could take O Φ • to consist of all those objects whose corepresented functor isisomorphic to one of the Φ i .The second choice is the ‘morally correct one,’ whereas the first choice turnsout to be more useful in our concrete applications. Remark . There is also a ‘coordinate free’ version of the above: namely,by the enriched Yoneda Lemma, O Φ • is isomorphic to the opposite of the fullsubcategory of Fun( C , SSet ) spanned by the Φ i . However, in our applications aswell as in our proofs, the above perspective will be more useful. Moreover, theabove is more akin to the classical Elmendorf Theorem where the orbit categoryis usually taken to be the full subcategory of transitive G -sets (instead of the fullsubcategory spanned by the fixed point functors themselves, viewed as simpliciallyenriched functors).Let us now fix a cellular family (Φ i ) i ∈ I and an orbit category O Φ • for it. Construction . We define the enriched functor Φ : C → Fun ( O opΦ • , SSet )as the composition C enriched Yoneda −−−−−−−−−−→ Fun ( C op , SSet ) restriction −−−−−−→ Fun ( O opΦ • , SSet ) . In other words, Φ( X )( Y ) = maps C ( Y, X ) with the obvious functoriality. In partic-ular, if Y ∈ O Φ • corepresents Φ i , then we have an isomorphism(1.1.12) ev Y ◦ Φ ∼ = Φ i as simplicially enriched functors. Lemma . (1) Φ preserves tensors. (2) Φ preserves filtered colimits. (3) For any i ∈ I , any X i corepresenting Φ i , and any n ≥ , the functor Φ preserves pushouts along ∂ ∆ n ⊗ X i ֒ → ∆ n ⊗ X i . Proof.
All of these claims can be checked levelwise. Because of the isomor-phism (1 . . (cid:3) Construction . It is well-known—see e.g. [
Kel05 , Theorem 4.51] to-gether with [
Kel05 , Theorem 3.73 op ] for a statement in much greater generality—that for any essentially small simplicial category T , any cocomplete category C en-riched and tensored over SSet and any simplicially enriched functor F : T → SSet , .1. EQUIVARIANT MODEL STRUCTURES AND ELMENDORF’S THEOREM 23 there exists an induced simplicial adjunction Fun ( T op , SSet ) ⇄ C with right ad-joint R given by R ( Y )( t ) = maps( F ( t ) , Y ) for all t ∈ T , Y ∈ C with the obviousfunctoriality in each variable.The left adjoint L can be computed by the simplicially enriched coend L ( X ) = Z t ∈ T X ( t ) ⊗ F ( t )for any enriched presheaf X , together with the evident functoriality.In particular, applying this to the inclusion O Φ • yields: Corollary . The simplicially enriched functor Φ has a simplicial leftadjoint Λ . (cid:3) Proposition . Let X ∈ O Φ • ⊂ C . Then the diagram (1.1.13) SSet Fun ( O opΦ • , SSet ) SSet C = – ⊗ Φ( X ) Λ– ⊗ X commutes up to canonical natural isomorphism. Moreover, the unit η : id ⇒ ΦΛ isan isomorphism on each object of the form K ⊗ Φ( X ) with K ∈ SSet . Proof.
Let us consider the diagram(1.1.14)
SSet Fun ( O opΦ • , SSet ) SSet C maps(Φ X, –)= ⇒ maps( X, –) Φ where the natural transformation is given on an object Y ∈ C by(1.1.15) Φ : maps( X, Y ) → maps(Φ X, Φ Y ) . As Φ( X ) = maps C (– , X ) | O Φ • = maps O opΦ • ( X, –), the enriched Yoneda Lemma im-plies that evaluation at id X ∈ (Φ X )( X ) defines an isomorphism maps(Φ X, Φ Y ) ∼ =Φ( Y )( X ) = maps( X, Y ). This is easily seen to be left inverse to (1 . . . .
14) is in fact an isomorphism. Accordingly, itstotal mate is again an isomorphism and this fills (1 . .
13) by construction.On the other hand, we can form the canonical mate of (1 . .
14) with respect tothe horizontal arrows. This is by definition the diagram
SSet Fun ( O opΦ • , SSet ) SSet C ⇒ – ⊗ Φ( X )= – ⊗ X Φ filled with the canonical comparison maps K ⊗ Φ( X ) → Φ( K ⊗ X ), K ∈ SSet ,which are isomorphisms by Lemma 1.1.27-(1). G -GLOBAL HOMOTOPY THEORY But by the calculus of mates this is equivalently the canonical mate of (1 . . SSet Fun ( O opΦ • , SSet ) SSet C Fun ( O opΦ • , SSet ) . = – ⊗ Φ( X ) ⇒ Λ =– ⊗ X Φ η ⇒ Since by the above both the transformation populating the left hand square as wellas the pasting are natural isomorphisms, we immediately conclude that η K ⊗ Φ( X ) isan isomorphism for all K ∈ SSet , finishing the proof. (cid:3)
Now we can state our version of Elmendorf’s Theorem:
Theorem . Let C be a complete and cocomplete category that is in ad-dition enriched, tensored, and cotensored over SSet , and let (Φ i ) i ∈ I be a cellularfamily on it. Then there exists a (necessarily unique) model structure on C inwhich a map f is a weak equivalence or fibration if and only if each Φ i ( f ) is a weakequivalence or fibration, respectively, in the Kan-Quillen model structure on SSet .This model structure is cofibrantly generated, simplicial, proper, and filteredcolimits in it are homotopical. If we choose for each i ∈ I an X i ∈ C corepresenting Φ i , then a set of generating cofibrations is given by (1.1.16) { ∂ ∆ n ⊗ X i ֒ → ∆ n ⊗ X i : i ∈ I, n ≥ } and a set of generating acyclic cofibrations is given by (1.1.17) { Λ nk ⊗ X i ֒ → ∆ n ⊗ X i : i ∈ I, ≤ k ≤ n } . Moreover, if O Φ • is any orbit category for the family Φ • , then the simplicial ad-junction (1.1.18) Λ : Fun ( O opΦ • , SSet ) ⇄ C : Φ is a Quillen equivalence (with respect to the projective model structure on the lefthand side). Finally, a commutative square (1.1.19) A BC D is a homotopy pushout in this model structure if and only if for each i ∈ I theinduced square Φ i ( A ) Φ i ( B )Φ i ( C ) Φ i ( D ) is a homotopy pushout in SSet . Proof.
We recall that the projective model structure on
Fun ( O opΦ • , SSet ) iscombinatorial, simplicial, proper, and that filtered colimits in it are homotopical(as they are in
SSet ). We will now verify the assumptions of Proposition 1.1.14 forthe adjunction (1 . . i ∈ I an Y i ∈ O Φ • corepresenting Φ i on C , then these cover all isomorphism classes in .1. EQUIVARIANT MODEL STRUCTURES AND ELMENDORF’S THEOREM 25 O Φ • . Thus, a set of generating cofibrations for the projective model structure on Fun ( O opΦ • , SSet ) is given by(1.1.20) { ∂ ∆ n ⊗ Φ( Y i ) ֒ → ∆ n ⊗ Φ( Y i ) : i ∈ I, n ≥ } whereas a set of generating acyclic cofibrations is given by(1.1.21) { Λ nk ⊗ Φ( Y i ) ֒ → ∆ n ⊗ Φ( Y i ) : i ∈ I, ≤ k ≤ n } , where we have again used that by definitionΦ( Y i ) = maps C (– , Y i ) | O Φ • = maps O opΦ • ( Y i , –) . We immediately see that for this choice the sources of the generating acyclic cofi-brations are themselves cofibrant, verifying Condition (1) of our criterion. More-over, the second half of Proposition 1.1.30 tells us that the unit is a weak equiv-alence (in fact, even an isomorphism) for every source or target of one of theabove generating cofibrations as well as on the initial object, verifying Condi-tion (2). Lemma 1.1.27-(2) verifies Condition (4 ′ ). For any generating cofibra-tion ∂ ∆ n ⊗ Φ( Y i ) ֒ → ∆ n ⊗ Φ( Y i ), Proposition 1.1.30 shows that its image underΛ agrees up to conjugation by isomorphisms with ∂ ∆ n ⊗ Y i ֒ → ∆ n ⊗ Y i , henceLemma 1.1.27-(3) implies Condition (3).Proposition 1.1.14 therefore applies to yield a model structure on C in which amap f is a weak equivalence respectively fibration if and only if Φ( f ) is a levelwiseweak equivalence respectively fibration, which in view of the isomorphism (1 . .
12) isthe desired model structure. The proposition also tells us that (1 . .
18) is a Quillenequivalence, that C is left proper and that filtered colimits in it are homotopical.Moreover, in conjunction with the first half of Proposition 1.1.30 it proves that apossible set of generating cofibrations is given by { ∂ ∆ n ⊗ Y i ֒ → ∆ n ⊗ Y i : i ∈ I, n ≥ } and a set of generating acyclic cofibrations by { Λ nk ⊗ Y i ֒ → ∆ n ⊗ Y i : i ∈ I, ≤ k ≤ n } . If now X i ∈ C is any other object corepresenting Φ i , then X i ∼ = Y i , hence also(1 . .
16) is a set of generating cofibrations and (1 . .
17) is a set of generating acycliccofibrations.Lemma 1.1.4 shows that C is also right proper and simplicial. It remains toprove the characterization of the homotopy pushouts. As homotopy pushouts in Fun ( O opΦ • , SSet ) (with respect to the injective and hence also with respect to theprojective model structure) are precisely the levelwise homotopy pushouts, thisfollows from Lemma 1.1.18 together with the isomorphism (1 . . (cid:3) We now want to apply the above ma-chinery in order to produce equivariant model structures for a simplicial monoid M . For this let us fix such M as well as a collection F of finite subgroups of M . Remark . As a simplicial category M -SSet agrees with the enrichedfunctor category Fun ( BM,
SSet ). Let us make the resulting enrichment, tensoring,and cotensoring explicit: • The tensoring is given by the categorical product where we consider ordi-nary simplicial sets as M -simplicial sets with trivial M -action. G -GLOBAL HOMOTOPY THEORY • The morphism space between
X, Y ∈ M -SSet is the simplicial subsetmaps M ( X, Y ) ⊂ maps( X, Y ) with n -simplices those X × ∆ n → Y thatare M -equivariant (with respect to the trivial M -action on ∆ n ). • The cotensoring is given by the underlying morphism spaces in
SSet ,equipped with the induced M -action. Lemma . For any G ∈ F , the enriched functor (–) G : M -SSet → SSet is corepresented by
M/G via evaluation at the class of ∈ M . It preserves fil-tered colimits, tensoring with SSet , and pushouts along underlying cofibrations. Inparticular, the family (cid:0) (–) G (cid:1) G ∈F is cellular. Proof.
The corepresentability statement is obvious. Since, limits and colimitsin M -SSet are created in SSet , filtered colimits commute with all finite limits, andhence in particular with fixed points with respect to finite groups. Similarly, onereduces the statement about pushouts to the corresponding statement in
Set , whichis trivial. Finally, the statement about tensoring is immediate from the concretedescription given in the previous remark.As for each X i corepresenting Φ i and any n ≥ ∂ ∆ n ⊗ X i → ∆ n ⊗ X i is in particular an underlying cofibration, the above in particular implies that thefamily of fixed point functors is cellular, finishing the proof. (cid:3) Definition . The orbit category O M, F is the full subcategory of M -SSet spanned by the M/G for G ∈ F . If F is clear from the context, we will also simplydenote the orbit category by O M . Remark . It will become convenient later that we define O M to consistonly of the M -simplicial sets M/G and not more generally those isomorphic tothem: namely, we remark that for any X ∈ O M there is a unique G such that X = M/G (equality as opposed to isomorphism); indeed, G can be recovered asthe class of 1 ∈ M in X . In particular, we can without ambiguity refer to an objectof O M as M/G and then do some constructions depending on G .Now we immediately get: Proof of Proposition 1.1.1.
By Lemma 1.1.33 the family (cid:0) (–) G (cid:1) G ∈F is cel-lular, so we may apply Theorem 1.1.31. This yields the desired model structure andverifies all the additional properties except M -SSet being combinatorial, which inturn follows immediately as the model structure is cofibrantly generated and since M -SSet is locally presentable as an ordinary category. (cid:3) In fact, Theorem 1.1.31 also immediately implies the generalization of the usualElemendorf Theorem to simplicial monoids:
Proposition . We have a simplicial Quillen equivalence
Λ :
Fun ( O op M , SSet ) ⇄ M -SSet : Φ where Φ( X )( M/G ) = maps M ( M/G, X ) ∼ = X G (with the evident functoriality ineach variable). (cid:3) Remark . We have cast the above proof in a way that relies heavily onsimplicial sets as this most easily integrates with the rest of this article. However, weremark that using essentially the same ideas one can also prove the correspondingstatement for topological spaces with the action of a given topological monoid, .1. EQUIVARIANT MODEL STRUCTURES AND ELMENDORF’S THEOREM 27 also cf. [
Ste16 , Corollary 3.20] for the case of topological groups and [
Sch20 ,Theorem 2.5] for a specific topological monoid.For a (finite) group G , it is classical that the cofibrations of the model structureassociated to the collection of all subgroups are precisely the underlying cofibra-tions. In the case of general simplicial monoids or if F is smaller, this will of courseusually fail horribly.For later use, we will now introduce an injective model structure that solvesthis issue, for which we will need: Corollary . Pushouts in M -SSet along underlying cofibrations arehomotopy pushouts (for any collection F of finite subgroups of M ). Proof.
By Lemma 1.1.33, each (–) G sends such a pushout to a pushout again.As taking fixed points moreover obviously preserves underlying cofibrations, this isthen a homotopy pushout in SSet , so the claim follows from Theorem 1.1.31. (cid:3)
Corollary . There is a (necessarily unique) model structure on M -SSet in which a map f is • a weak equivalence if and only if f G is a weak homotopy equivalence forall G ∈ F , and • a cofibration if and only if it is an underlying cofibration (i.e. levelwiseinjective).We call this the injective F -model structure (or equivariant injective model struc-ture if F is clear from the context). It is combinatorial, simplicial, proper, andfiltered colimits in it are homotopical. Proof.
As an ordinary category, M -SSet is just a simplicially enriched func-tor category, and hence the usual injective model structure (which has as weakequivalences the underlying non-equivariant weak homotopy equivalences!) on itexists and is combinatorial. On the other hand, the F -weak equivalences are stableunder pushouts along arbitrary underlying cofibrations by the previous corollary.Hence, we can apply Corollary A.2.11 which provides the desired model structureand proves that it is combinatorial, proper, and that filtered colimits in it arehomotopical.It only remains to prove that it is simplicial, which means verifying the pushout-product axiom. So let i : K ֒ → L be a cofibration of simplicial sets and let f : X → Y be an underlying cofibration of M -simplicial sets. Because SSet is a simplicialmodel category, we immediately see that the pushout product map K × X K × YL × X ( K × Y ) ∐ K × X ( L × X ) L × Y i × X K × f i × YL × f i (cid:3) f is again an underlying cofibration. It only remains to prove that this is a weakequivalence provided that either i or f is. For this we observe that the equivariantweak equivalences are stable under finite products by Lemma 1.1.2; moreover, aweak homotopy equivalence between simplicial sets with trivial M -action is already G -GLOBAL HOMOTOPY THEORY an equivariant weak equivalence. Hence, if i is an acyclic cofibrations of simplicialsets, then the cofibration i × X is actually acyclic in the equivariant injective modelstructure and so is i × Y . Moreover, K × Y → ( K × Y ) ∐ K × X ( L × X ) is also anacyclic cofibration as the pushout of an acyclic cofibration. It follows by 2-out-of-3that also i (cid:3) f is an equivariant weak equivalence. The argument for the case that f is an acyclic cofibration is analogous, and this finishes the proof. (cid:3) We close this section by explaining how the modelstructures constructed above for different monoids and families relate to each other.1.1.4.1.
Change of monoid. If α : H → G is any group homomorphism, then α ∗ obviously preserves cofibrations, fibrations, and weak equivalences of the A ℓℓ -model structures. It follows immediately that the adjunctions α ! ⊣ α ∗ and α ∗ ⊣ α ∗ are Quillen adjunctions. For monoids and general F one instead has to distinguishbetween the usual F -model structure and the injective one: Lemma . Let α : M → N be any monoid homomorphism, let F be acollection of finite subgroups of M , and let F ′ be a collection of finite subgroupsof N such that α ( G ) ∈ F ′ for all G ∈ F . Then α ∗ sends F ′ -weak equivalences to F -weak equivalences and it is part of a simplicial Quillen adjunction α ! : M -SSet F -equivariant ⇄ N -SSet F ′ -equivariant : α ∗ . Proof. If f is any morphism in N -SSet and G ⊂ M is any subgroup, then( α ∗ f ) G = f α ( G ) . Thus, the claim follows immediately from the definition of theweak equivalences and fibrations of the equivariant model structures. (cid:3) Lemma . In the situation of the above lemma, also α ∗ : N -SSet F ′ -equivariant injective ⇄ M -SSet F -equivariant injective : α ∗ . is a simplicial Quillen adjunction. Proof.
We have seen in the previous lemma that α ∗ is homotopical. Moreover,it obviously preserves injective cofibrations. (cid:3) The question whether α ! is left Quillen for the injective model structures—orwhether α ∗ is right Quillen for the usual model structures—with respect to suitablefamilies F , F ′ is more complicated. The following propositions will cover the casesof interest to us: Proposition . Let α : H → G be an injective homomorphism of discretegroups and let M be any simplicial monoid. Let F be any collection of finite sub-groups of M × H and let F ′ be a collection of finite subgroups of M × G such thatthe following holds: for any K ∈ F ′ , g ∈ G also ( M × α ) − ( gKg − ) ∈ F . Then α ! : ( M × H )-SSet F -equiv. inj. ⇄ ( M × G )-SSet F ′ -equiv. inj. : α ∗ = ( M × α ) ∗ is a simplical Quillen adjunction; in particular, α ! is homotopical. Proof.
The proof relies on an explicit description of the left adjoint availablein this special case. Namely, we may assume without loss of generality that H isa subgroup of G and α is its inclusion. Then G × H – (with the M -action pulledthrough via enriched functoriality) is a model for α ! ; if X is any H -simplicial set,then the n -simplices of G × H X are of the form [ g, x ] with g ∈ G and x ∈ X n where[ g, x ] = [ g ′ , x ′ ] if and only if there exists an h ∈ H with g ′ = gh and x = h.x ′ . .1. EQUIVARIANT MODEL STRUCTURES AND ELMENDORF’S THEOREM 29 Now let K ⊂ G × M be any subgroup. We set S := { g ∈ G : g − Kg ⊂ H × M } and observe that this is a right H -subset of G . We fix representatives s , . . . , s r ofthe orbits. If now X is any ( H × M )-simplicial set, then we define ι : r a i =1 X s − i Ks i → G × H X as the map that is given on the i -th summand by x [ s i , x ]. Claim.
The map ι is natural in X (with respect to the evident functorialityon the left hand side) and it defines an isomorphism onto ( G × H X ) K . Proof.
The naturality part is obvious. Moreover, it is clear from the choice ofthe s i as well as the above description of the equivalence relation that ι is injective,so that it only remains to prove that its image equals ( G × H X ) K .Indeed, assume [ g, x ] is a K -fixed n -simplex. Then in particular k g ∈ gH forany k = ( k , k ) ∈ K by the above description of the equivalence relation, hence g − k g ∈ H which is equivalent to g − kg ∈ H × M . Letting k vary, we concludethat g ∈ S and after changing the representative if necessary we may assume that g = s i for some 1 ≤ i ≤ r . But then[ s i , x ] = k. [ s i , x ] = [ k s i , k .x ] = [ s i ( s − i k s i ) , k .x ]= [ s i , ( s − i k s i , k ) .x ] = [ s i , ( s − i ks i ) .x ]for any k ∈ K , and hence x ∈ X s − i Ks i . Thus, im ι contains all K -fixed points. Onthe other hand, going through the above equation backwards shows that [ s i , x ] is K -fixed for any ( s − i Ks i )-fixed x , i.e. also im ι ⊂ ( G × H X ) K . △ In particular, for K = 1 this recovers the fact that non-equivariantly G × H X is given as disjoint union of copies of X ; we immediately conclude that α ! preservesinjective cofibrations. On the other hand, if K ∈ F ′ then we conclude from theclaim that for any morphism f in ( H × M )-SSet the map ( G × H f ) K is conjugateto ` ri =0 f s − i Ks i for some s , . . . , s r ∈ G with s − i Ks i ⊂ H × M for all i = 0 , . . . , r .Then by assumptions on F already s − i Ks i ∈ F , so that each f s − i Ks i is a weakhomotopy equivalence whenever f is a F -weak equivalence. As coproducts of sim-plicial sets are fully homotopical, we conclude that ( G × H f ) K is a weak homotopyequivalence, and letting K vary this shows G × H f is a F ′ -weak equivalence. (cid:3) Proposition . In the situation of the previous proposition also the sim-plicial adjunction α ∗ : ( M × G )-SSet F ′ -equivariant ⇄ ( M × H )-SSet F -equivariant : α ∗ . is a Quillen adjunction. Moreover, if ( G : im α ) < ∞ , then α ∗ is fully homotopical. Proof.
We may again assume that α is the inclusion of a subgroup, so that α ∗ can be modelled by maps H ( G, –).Let K ⊂ M × G be any subgroup, and let K be its projection to G . We pick asystem of representatives ( g i ) i ∈ I of H \ G/K and we let L i = ( M × H ) ∩ ( g i Kg − i ).Similarly to the previous proposition one checks that we have an isomorphism ` i ∈ I ( M × H ) /L i → ( M × G ) /K given on summand i by [ m, h ] [ m, hg i ]. To-gether with the canonical isomorphism maps H ( G, –) K ∼ = maps M × H (( M × G ) /K, –)induced by the projection, this shows that for any ( M × H )-equivariant map G -GLOBAL HOMOTOPY THEORY f : X → Y the map α ∗ ( f ) K is conjugate to Q i ∈ I f L i . If K ∈ F ′ , then the as-sumptions guarantee that L i ∈ F , so α ∗ is obviously right Quillen. If in addition( G : H ) < ∞ , then H \ G is finite, and hence so is I . As finite products in SSet arehomotopical, we conclude that also α ∗ is in this case. (cid:3) Remark . Let
A, B be groups. We recall that a graph subgroup C ⊂ A × B is a subgroup of the form { ( a, ϕ ( a )) : a ∈ A ′ } for some subgroup A ′ ⊂ A and some group homomorphism ϕ : A ′ → B ; note that this is not symmetric and A and B . Both A ′ and ϕ are actually uniquely determined by C , and we write C =: Γ A ′ ,ϕ . In general, a subgroup C ⊂ A × B is a graph subgroup if and only if C ∩ B = 1.If A and B are monoids, then we can define its graph subgroups as the graphsubgroups of the maximal subgroup core( A × B ) of A × B . If A ′ ⊂ core( A ) and ϕ : A ′ → B is a homomorphism, then we will abbreviate (–) ϕ := (–) Γ A ′ ,ϕ .Now let E be any collection of finite subgroups of M closed under taking sub-conjugates. Then the assumptions of the previous two propositions are in particularsatisfied if we take F = G E ,H to be the collection of those graph subgroups Γ K,ϕ of M × H with K ∈ E , and similarly F ′ = G E ,G .Let us consider a general homomorphism α : H → G now. Then ( M × α ) ∗ isright Quillen with respect to the G E ,H - and G E ,G -model structures for all E , so KenBrown’s Lemma implies that α ! preserves weak equivalences between cofibrant ob-jects. On the other hand, the above proposition implies that α ! is fully homotopicalif α is injective. The following proposition interpolates between these two results: Proposition . Let E be a collection of finite subgroups of M closedunder subconjugates, let α : H → G be a homomorphism, and let f : X → Y bea G E ,H -weak equivalence in ( M × H )-SSet such that ker α acts freely on both X and Y . Then α ! ( f ) is a G E ,G -weak equivalence. Proof.
By Proposition 1.1.42 we may assume without loss of generality that α is the quotient map H → H/ ker( α ), so that the functor α ! can be modelled byquotiening out the action of the normal subgroup K := ker( α ).The following splitting follows from a simple calculation similar to the abovearguments, which we omit. It can also be obtained from the discrete special caseof [ Hau17 , Lemma A.1] by adding disjoint basepoints:
Claim.
Let L ⊂ M be any subgroup and let ϕ : L → G/K be any homomor-phism. Then we have for any ( L × G )-simplicial set Z on which K acts freely anatural isomorphism a [ ψ : L → G ] Z ψ / (C G (im ψ ) ∩ K ) ∼ = −→ ( Z/K ) ϕ given on each summand by [ z ] [ z ]. Here the coproduct runs over K -conjugacyclasses of homomorphisms lifting ϕ , and C G denotes the centralizer in G . △ We can now prove the proposition. Let L ∈ E and let ϕ : L → G . In orderto show that ( f /K ) ϕ is a weak homotopy equivalence it suffices by the claim that f ψ / (C G (im ψ ) ∩ K ) be a weak homotopy equivalence for all ψ : L → G lifting ϕ . Butindeed, as K acts freely on X and Y , so does C G (im ψ ) ∩ K ; in particular, it also actsfreely on X ψ and Y ψ . The claim follows because f ψ is a weak homotopy equivalence .2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 31 by assumption and because free quotients preserve weak homotopy equivalences ofsimplicial sets (e.g. by the special case M = 1 of the above discussion). (cid:3) Change of family.
We now turn to the special case that the monoid M is fixed (i.e. α = id M ), but the family F is allowed to vary. For this we will usethe notion of quasi-localizations , which we recall in Appendix A.1; in particular, wewill use from there the notation C ∞ W for ‘the’ quasi-localization of a category C ata class W of maps. Proposition . Let M be a simplicial monoid and let F , F ′ be collectionsof finite subgroups of M such that F ′ ⊂ F . Then the identity descends to a quasi-localization (1.1.22) M -SSet ∞F -weak equivalences → M -SSet ∞F ′ -weak equivalences at the F ′ -weak equivalences, and this functor admits both a left adjoint λ as well asa right adjoint ρ . Both λ and ρ are fully faithful. Proof.
The identity obviously descends to the quasi-localization (1 . . Cis19 , Proposition 7.1.17].But indeed, Lemma 1.1.40 specializes to yield a Quillen adjunctionid : M -SSet F ′ -equivariant ⇄ M -SSet F -equivariant : idso that the left derived functor L id in the sense of Theorem A.1.19 defines thedesired left adjoint. To construct the right adjoint, we observe that whileid : M -SSet F -equivariant ⇄ M -SSet F ′ -equivariant : idis in general not a Quillen adjunction with respect to the usual model structures, itbecomes one if we use Corollary A.2.10 to enlarge the cofibrations on the right handside to contain all generating cofibrations of the F -model structure (which we areallowed to do by Corollary 1.1.38), or alternatively that it is a Quillen adjunctionfor the corresponding injective model structures by Lemma 1.1.41. (cid:3) Remark . By the above proof, λ can be modelled by taking a cofibrantreplacement with respect to the F ′ -model structure. G -global homotopy theory via monoid actions1.2.1. The universal finite group. Schwede [
Sch20 ] proved that unstableglobal homotopy theory with respect to all compact Lie groups can be modelled asspaces with the action of a certain topological monoid L that he calls the universalcompact Lie group , and which will be recalled in Section 1.5.For unstable global homotopy theory with respect to finite groups that, we willbe interested in a certain (discrete) monoid M , which is also crucial to Schwede’sapproach [ Sch19b ] to global algebraic K -theory. Definition . We write ω = { , , . . . } and we denote by M the monoid(under composition) of all injections ω → ω .Analogously to the Lie group situation [ Sch20 , Definition 1.4], when we modela global space by an M -equivariant space X , we do not expect all the fixed pointspaces X G for G ⊂ M to carry homotopical information, but only for certain G -GLOBAL HOMOTOPY THEORY so-called universal G . To define these, we first need the following terminology,cf. [ Sch19b , Definition 2.16]:
Definition . Let G be any finite group. A countable G -set X is called a complete G -set universe if some (hence any) of the following equivalent conditionsholds:(1) Any finite G -set embeds G -equivariantly into X .(2) Any countable G -set embeds G -equivariantly into X .(3) There exists a G -equivariant isomorphism X ∼ = ∞ a i =0 a H ⊂ G subgroup G/H. (4) Any subgroup H ⊂ G occurs as stabilizer of infinitely many distinct ele-ments of X .The proof that the above conditions are indeed equivalent is easy and we omitit. For all of these statements except for the second one this also appears withoutproof as [ Sch19b , Proposition 2.17 and Example 2.18].The following lemmas are similarly straightforward to prove from the definitionsand they also appear without proof as part of [
Sch19b , Proposition 2.17].
Lemma . Let X ⊂ Y be G -sets, assume X is a complete G -set universeand Y is countable. Then also Y is a complete G -set universe. (cid:3) Lemma . Let X be a complete G -set universe and let α : H → G be aninjective group homomorphism. Then α ∗ X (i.e. X with H -action given by h.x = α ( h ) .x ) is a complete H -set universe. (cid:3) Definition . A finite subgroup G ⊂ M is called universal if the re-striction of the tautological M -action on ω to G makes ω into a complete G -setuniverse.Lemma 1.2.4 immediately implies: Corollary . Let H ⊂ G ⊂ M be subgroups and assume that G is uni-versal. Then also H is universal. (cid:3) Lemma . Let G be any finite group. Then there exists an injective monoidhomomorphism i : G → M with universal image. Moreover, if j : G → M isanother such homomorphism, then there exists a ϕ ∈ core M such that (1.2.1) j ( g ) = ϕi ( g ) ϕ − for all g ∈ G . Proof.
This is similar to [
Sch20 , Proposition 1.5]: the complete G -set uni-verse U := ∞ a i =0 a H ⊂ G subgroup G/H. is countable, so after picking a bijection of sets ω ∼ = U we get a G -action on ω turning it into a complete G -set universe. The G -action amounts to an injection G → Σ ω ⊂ M which is then the desired homomorphism i . .2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 33 If j is another such homomorphism, then both i ∗ ω and j ∗ ω are complete G -set universe and hence there exists a G -equivariant isomorphism ϕ : i ∗ ω → j ∗ ω ,see e.g. [ Sch19b , Proposition 2.17-(ii)] where this appears without proof. The G -equivariance then precisely means that ϕ ( i ( g )( x )) = j ( g )( ϕ ( x )) for all g ∈ G and x ∈ ω , i.e. ϕi ( g ) ϕ − = j ( g ) as desired. (cid:3) Heuristically, we would like to think of a global space X as having for each abstract finite group G a fixed point space X G and for each abstract group ho-momorphism f : G → H a restriction map f ∗ : X H → X G . One way to makethis heuristic rigorous is given by Schwede’s orbispace model, see [ Sch19a , Theo-rem 2.12] (in fact, it turns out that there is also some additional 2-functoriality).The above lemma already tells us that we can to an M -space assign, for each ab-stract finite group G , an essentially unique fixed point space X G as follows: wepick an injective group homomorphism i : G → M with universal image and set X G := X i ( G ) . This is indeed independent of the group homomorphism i up to iso-morphism: namely, if j is another such group homomorphism, the lemma providesus with a ϕ such that j ( g ) = ϕi ( g ) ϕ − for all g ∈ G and is easy to check that ϕ. – : X → X restricts to an isomorphism X i ( G ) → X j ( G ) .However, there is an issue here—namely, the element ϕ (or more precisely, itsaction) is not canonical (and hence it is not clear how to define 1-functoriality): Example . Let us consider the special case G = 1, so that there is inparticular only one homomorphism i = j : G → M . Then any ϕ ∈ core M satisfiesthe condition (1 . .
1) and hence produces an endomorphism ϕ. – of X = X { } .Looking at the orbispace model, we should expect X { } to only come with(homotopically) trivial endomorphisms. However, for X = M any ϕ = 1 gives usa non-trivial endomorphism. The above suggests that M -simplicial setswith respect to maps inducing weak homotopy equivalences on fixed points foruniversal subgroups are not yet a model of unstable global homotopy theory. Inorder to solve the issue raised in the example, we will enhance M to a simplicialor categorical monoid in particular trivializing the conjugation action. This uses: Construction . Let X be any set. We write EX for the (small) categorywith objects X and precisely one morphism x → y for each x, y ∈ X , which wedenote by ( y, x ). We extend E to a functor Sets → Cat in the obvious way.We will moreover also write EX for the simplicial set given in degree n by( EX ) n = X × (1+ n ) ∼ = maps( { , . . . , n } , X )with structure maps via restriction and with the evident functoriality in X . Weremark that the simplicial set EX is indeed canonically isomorphic to the nerveof the category EX , justifying the clash of notation. Indeed, it will be useful atseveral points to switch between viewing EX as a category or as a simplicial set. Remark . It is clear that the category EX is a groupoid and it is con-tractible for X = ∅ . In particular, the simplicial set EX is a Kan complex, againcontractible unless X = ∅ .The functor E : Sets → Cat is right adjoint to the functor Ob :
Cat → Sets ;likewise E : Sets → SSet is right adjoint to the functor sending a simplicial set toits set of zero simplices. In particular, E preserves products, so E M is canonically G -GLOBAL HOMOTOPY THEORY a simplicial monoid. As it is contractible, any two translations u. – , v. – for u, v ∈ M are homotopic on any E M -object C ; in fact, there is unique edge ( v, u ) from v to u in E M , and acting with this gives an explicit homotopy u. – ⇒ v. –.We conclude that E M avoids the issue detailed in Example 1.2.8. Indeed,Theorem 1.3.30 together with Theorem 1.5.23 will show that E M -SSet is a modelof global homotopy theory. Maybe somewhat surprisingly, the main result of thissubsection (Theorem 1.2.20) will be that the same homotopy theory can be modelledby M -simplicial sets with respect to a slightly intricate notion of weak equivalence.Before we will introduce G -global model categories based on E M - and M -actions in the next subsection, let us first consider the ordinary global situation inorder to present the main ideas without being overly technical. The basis for thiswill be the following model categories provided by Proposition 1.1.1: Corollary . The category M -SSet admits a unique model structuresuch that a map f : X → Y is a weak equivalence or fibration if and only if for eachuniversal G ⊂ M the map f G is a weak homotopy equivalence or Kan fibration,respectively. We call this the universal model structure and its weak equivalencesthe universal weak equivalences .This model structure is simplicial, combinatorial, and proper. A possible set ofgenerating cofibrations is given by I = {M /G × ∂ ∆ n ֒ → M /G × ∆ n : n ≥ , G universal } and a possible set of generating acyclic cofibrations is given by J = {M /G × Λ nk ֒ → M /G × ∆ n : 0 ≤ k ≤ n, G universal } . Moreover, filtered colimits are homotopical in M -SSet . (cid:3) Remark . As already mentioned above, the above will not yet modelglobal equivariant homotopy theory, so we reserve the names ‘global model struc-ture’ and ‘global weak equivalences’ for a different model structure.
Corollary . The category E M -SSet of E M -simplicial sets admits aunique model structure such that a map f : X → Y is a weak equivalence or fibrationif and only if it is so when considered as a map in M -SSet . We call this the globalmodel structure and its weak equivalences the global weak equivalences .This model structure is simplicial, combinatorial, and proper. A possible set ofgenerating cofibrations is given by I = { ( E M ) /G × ∂ ∆ n ֒ → ( E M ) /G × ∆ n : n ≥ , G universal } and a possible set of generating acyclic cofibrations is given by J = { ( E M ) /G × Λ nk ֒ → ( E M ) /G × ∆ n : 0 ≤ k ≤ n, G universal } . Moreover, filtered colimits are homotopical in E M -SSet . (cid:3) Construction . The forgetful functor E M -SSet → M -SSet admitsboth a left and a right adjoint. While they exist for abstract reasons (either by theSpecial Adjoint Functor Theorem or as simplicially enriched Kan extensions), theyare also easy to make explicit:Let X be any M -simplicial set. We write E M × M X for the following E M -simplicial set: as a simplicial set, this is the quotient of E M × X under the equiv-alence relation generated in degree n by ( m u, . . . , m n u ; x ) ∼ ( m , . . . , m n ; ux ) for .2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 35 m , . . . , m n , u ∈ M and x ∈ X n . As usual we denote the class of ( m , . . . , m n ; x )by [ m , . . . , m n ; x ].The E M -action on E M × M X is induced by the obvious E M -action on thefirst factor. If f : X → Y is any M -equivariant map, then E M × M f is inducedby E M × f . We omit the easy verification that is well-defined.We have a natural M -equivariant map η : X → forget( E M × M X ) given indegree n by sending an n -simplex x to the class [1 , . . . , x ]. Moreover, if Y is an E M -simplicial set, we have an E M -equivariant map ǫ : E M × M (forget Y ) → Y giving in degree n by acting, i.e. [ m , . . . , m n ; y ] ( m , . . . , m n ) .y . We leave theeasy verification to the reader that these are well-defined, natural, and define a unitrespectively counit for the claimed adjunction E M × M – ⊣ forget.Similarly, the forgetful functor admits a right adjoint maps M ( E M , –), wherethe E M -action is induced by the right E M -action on itself via postcomposition.By definition of the model structures we immediately get: Corollary . The simplicial adjunction (1.2.2) E M × M – : M -SSet ⇄ E M -SSet : forget is a Quillen adjunction and the right adjoint creates weak equivalences. (cid:3) As already suggested by our heuristic above, this is not a Quillen equivalence;more precisely, forget ∞ is not essentially surjective: Example . If X is any E M -simplicial set, then all u ∈ M act on X byweak homotopy equivalences, and hence the same will be true for any M -simplicialset Y weakly equivalent to forget X . On the other hand, M considered as a discrete M -simplicial set with M -action by postcomposition does not satisfy this, and hencecan’t lie in the essential image.In the example we have only looked at the underlying non-equivariant homotopytype of a given M -simplicial set. However, in order to have a sufficient criterionfor the essential image of forget we should better look at the equivariant homotopytype. While the translation maps will usually not be M -equivariant (related to thefact that M is highly non-commutative), we can look at the ‘maximal action’ thatwe can still expect to be preserved: Definition . A M -simplicial set X is called semistable if for any uni-versal subgroup G ⊂ M and any u ∈ M centralizing G the translation map u. – : X → X is a genuine G -weak equivalence.We observe that this is equivalent to demanding that for any such u ∈ M and G ⊂ M the restriction of u. – to X G → X G is a weak homotopy equivalence (thisuses that subgroups of universal subgroups are themselves universal). Remark . The term ‘semistable’ refers to Schwede’s characterization[
Sch08 , Theorem 4.1 and Lemma 2.3-(iii)] of semistable symmetric spectra (i.e. sym-metric spectra whose na¨ıve homotopy groups agree with their derived homotopygroups) as those spectra for which a certain canonical M -action on the na¨ıve homo-topy groups is given by isomorphisms, also cf. [ Hau17 , Corollary 3.32 and Propo-sition 3.16] for a similar characterization in the equivariant case due to Hausmann. G -GLOBAL HOMOTOPY THEORY Both Schwede and Hausmann prove that in the respective situation the actionon the na¨ıve homotopy groups is actually trivial (i.e. all elements of M act by theidentity on π ∗ ). Likewise, it will follow from Theorem 1.2.20 together with theproof of Lemma 1.2.19 below that for a semistable M -simplicial set the translation u. –, u ∈ M centralizing some universal subgroup G ⊂ M , is in fact the identity inthe G -equivariant homotopy category. Lemma . Let X be any E M -simplicial set. Then forget X is semistable. Proof.
While this will follow automatically from our comparison theorem, letus give a direct proof for motivational purposes:We show even stronger that u. – : X G → X G is G -equivariantly homotopic to theidentity for any subgroup G ⊂ M centralized by u . Indeed, an explicit equivarianthomotopy is given by acting with ( u, ∈ ( E M ) . (cid:3) Obviously, semistability is invariant under universal weak equivalences, so it isa necessary condition to lie in the essential image of forget by the lemma. As themain result of this section, we will show that it is also sufficient, and moreover theabove is everything that prevents forget ∞ from being an equivalence: Theorem . The adjunction (1 . . induces a Bousfield localization E M × L M – : M -SSet ∞ ⇄ E M -SSet ∞ : forget ∞ ; in particular, forget ∞ is fully faithful. Moreover, its essential image consists pre-cisely of the semistable objects. Remark . The theorem in particular tells us that the forgetful func-tor identifies E M -SSet ∞ with the full subcategory of M -SSet ∞ spanned by thesemistable objects. In view of Proposition A.1.15 the latter is canonically identi-fied with the quasi-localization of semistable M -simplicial sets at the same weakequivalences (so we do not have to be careful to distinguish them). In other words:semistable M -simplicial sets with respect to the universal weak equivalences are amodel of unstable global homotopy theory.Our proof of the theorem will proceed indirectly via the alternative modelsprovided by the Elmendorf Theorem for monoids (Proposition 1.1.36): namely, wewant to exhibit O E M as an explicit simplicial localization of O M in the senseof Definition A.1.5, and then deduce the theorem from the universal property ofquasi-localizations (or more precisely its model categorical manifestation Theo-rem A.1.16). To do so, let us begin by understanding the orbit categories a bitbetter: Lemma . Let ( u , . . . , u n ) ∈ M n +1 such that [ u , . . . , u n ] is an H -fixedelement of ( E M /G ) n . Then there exists for any h ∈ H a unique σ ( h ) such that hu i = u i σ ( h ) for all i . Moreover, σ : H → G is a group homomorphism.Conversely, whenever such a σ exists, [ u , . . . , u n ] is H -fixed in ( E M ) /G . Proof.
As [ u , . . . , u n ] is H -fixed, ( u , . . . , u n ) ∼ ( hu , . . . , hu n ), so by defini-tion there indeed exists some σ ( h ) ∈ G such that u i = u i σ ( h ) for all i ; moreover, σ ( h ) is unique as G acts freely from the right on M .To check that σ is a group homomorphism, let h , h ∈ H arbitrary. Then h h u = h uσ ( h ) = uσ ( h ) σ ( h ), hence σ ( h h ) = σ ( h ) σ ( h ) by uniqueness.The proof of the converse is trivial. (cid:3) .2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 37 Construction . By definition, O E M ⊂ E M -SSet is the full sim-plicial subcategory spanned by the ( E M ) /G for universal G ⊂ M . We haveseen in Lemma 1.1.33 that maps O E M ( E M /G, E M /H ) ∼ = ( E M /H ) G via evalu-ation at [1] ∈ E M /G . On 0-simplices, this gives maps O E M ( E M /G, E M /H ) ∼ =( M /H ) G ; an inverse is then given by sending u ∈ ( M /H ) G to – · u : [ m , . . . , m n ] [ m u, . . . , m n u ].More generally, any n -cell of the mapping space maps O E M ( E M /G, E M /H )can be represented by an ( n + 1)-tuple ( u , . . . , u n ) such that [ u , . . . , u n ] ∈ E M /H is G -fixed, which by Lemma 1.2.22 is equivalent to the existence of a group ho-momorphism σ : G → H such that gu i = u i σ ( g ) for all i = 0 , . . . , n . Two tuplesrepresent the same morphism iff they become equivalent in E M /H , i.e. iff theyonly differ by right multiplication with some h ∈ H .Moreover, one immediately sees by direct inspection, that if the n -cell f ∈ maps O E M ( E M /G, E M /H ) n is represented by ( u , . . . , u n ) and the n -cell f ′ ∈ maps O E M ( E M /H, E M /K ) n is represented by ( u ′ , . . . , u ′ n ), then their composition f ′ f is represented by ( u u ′ , . . . , u n u ′ n ) (observe the different order!).Likewise, O M is the full simplicial subcategory of M -SSet spanned by the M /G and we have Hom( M /G, M /H ) ∼ = ( M /H ) G via evaluation at [1]; composi-tion is again induced from multiplication in M .We now define a functor i : O M → O E M as follows: an object M /G is sent to E M /G . On morphism spaces, i is given as the compositionmaps O M ( M /G, M /H ) ∼ = ( M /H ) G ∼ = maps O E M ( E M /G, E M /H ) ֒ → maps O E M ( E M /G, E M /H ) , which then just sends the morphism represented by u to the morphism representedby the same element. As an upshot of the above discussion this is indeed functorial. Definition . Let G ⊂ M be universal. A map f : M /G → M /G iscalled centralizing if there exists a u ∈ M centralizing G such that f is given byright multiplication by u . Analogously, we define centralizing morphisms in O E M .The following will be the main ingredient to the proof of Theorem 1.2.20: Proposition . The functor i : O M → O E M is a simplicial localizationat the centralizing morphisms. Proof.
By construction, i induces an isomorphism onto the underlying cat-egory u O E M of O E M . Thus it is enough to prove that u O E M ֒ → O E M is aquasi-localization at the centralizing morphisms, for which we will check the con-ditions of Proposition A.1.10, i.e. that the centralizing morphisms are homotopyequivalences, and that the iterated degeneracies(1.2.3) s ∗ : (cid:0) ( O E M ) , W (cid:1) → (cid:0) ( O E M ) n , s ∗ W (cid:1) induce equivalences on quasi-localizations, where W denotes the class of centralizingmorphisms.For the first claim we note that if u centralizes G , then – · u is even homotopicto the identity via the edge [1 , u ] in ( E M /G ) G ∼ = maps O E M ( E M /G, E M /G ).For the second statement it is by Corollary A.1.14 enough to prove that (1 . . i : [0] → [ n ] , i ∗ is obviously G -GLOBAL HOMOTOPY THEORY homotopical and moreover i ∗ s ∗ = ( si ) ∗ = id by functoriality. We will now provethat s ∗ i ∗ is homotopic to the identity of ( O E M ) n .We begin by picking for each universal G ⊂ M a G -equivariant isomorphism ϕ G : ω ∐ ω ∼ = ω , where G acts on each of the three copies of ω in the tautologicalway; such an isomorphism indeed exists as both sides are complete G -set universes.Restricting ϕ G to the two copies of ω gives injections α G , β G : ω → ω such that:(1) α G and β G centralize G (2) ω = im( α G ) ⊔ im( β G ).We now define f : ( O E M ) n → ( O E M ) n as follows: f is the identity on objects. Amorphism ( E M ) /G → ( E M ) /H represented by ( u , . . . , u n ) ∈ M n +1 is sent tothe morphism represented by ( v , . . . , v n ) where v i satisfies(1.2.4) v i α H = α G u i and v i β H = β G u . We first observe that there is indeed a unique such v i as α H and β H are injectionswhose images form a partition of ω . Moreover, this is an injection, as α G and β G have disjoint image and as both α G u i and β G u are injective.Next, we show that ( v , . . . , v n ) indeed defines a morphism, i.e. it represents a G -fixed object of ( E M ) /H . Indeed, as ( u , . . . , u n ) represents a G -fixed object, wehave a (unique) group homomorphism σ : G → H such that gu i = u i σ ( g ) for all g ∈ G . But then gv i α H = gα G u i = α G gu i = α G u i σ ( g ) = v i α H σ ( g ) = v i σ ( g ) α H , where we have used Condition (1) twice as well as the definition of v i . Analogouslyone shows gv i β H = v i σ ( g ) β H ; as the images of α H and β H together cover ω weconclude that gv i = v i σ ( g ) for all i and all g and hence [ v , . . . , v n ] ∈ ( E M ) /H is G -fixed as desired.Moreover, this is independent of the choice of representative: if we pick anyother representative ( u ′ , . . . , u ′ n ), then there is some h ∈ H such that u ′ i = u i h forall i , and thus the associated v ′ i satisfy v ′ i α H = α G u ′ i = α G u i h = v i α H h = v i hα H , where we have used the definitions of v i and v ′ i as well as (1). Analogously oneshows v ′ i β H = v i hβ H ; as before we conclude that v ′ i = v i h , so that ( v , . . . , v n ) and( v ′ , . . . , v ′ n ) represent the same morphism.With this established, one easily checks that f is a functor ( O E M ) n → ( O E M ) n .It only remains to prove that f is homotopic to both the identity and s ∗ i ∗ . For thiswe observe that we have by the defining equation (1 . .
4) natural transformationsid ⇐ f ⇒ s ∗ i ∗ where the left hand transformation is given on ( E M ) /G by the morphism rep-resented by ( α G , . . . , α G ) while the right hand transformation is represented by( β G , . . . , β G ). Condition (1) ensures that these are levelwise weak equivalences,finishing the proof. (cid:3) Corollary . In the Quillen adjunction ( i op ) ! : Fun ( O op M , SSet ) ⇄ Fun ( O op E M , SSet ) : ( i op ) ∗ , the right adjoint is homotopical and the induced functor between associated quasi-categories is fully faithful with essential image precisely those simplicial presheaveson O M that invert the centralizing morphisms. .2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 39 Proof.
It is clear, that ( i op ) ∗ is homotopical. By the previous proposition, i : O M → O E M is a simplicial localization at the centralizing morphisms, and hence i op is a simplicial localization at their opposites. As both source and target of i op are small and locally fibrant, the claim now follows from Theorem A.1.16. (cid:3) Proof of Theorem 1.2.20.
We already know from Corollary 1.2.15 that E M × M (–) ⊣ forget is a Quillen adjunction with homotopical right adjoint. Ittherefore suffices to prove that forget ∞ is fully faithful with essential image thesemistable M -simplicial sets. Claim.
The diagram(1.2.5) E M -SSet M -SSetFun ( O op E M , SSet ) Fun ( O op M , SSet ) forgetΦ Φ( i op ) ∗ of homotopical functors commutes up to natural isomorphism. Proof.
An explicit choice of such an isomorphism τ is given as follows: if X isany E M -simplicial set and G ⊂ M is universal, then τ X ( M /G ) is the compositionΦ( X )( i ( M /G )) = Φ( X )( E M /G ) ev [1] −−−→ X G = (forget X ) G (ev [1] ) − −−−−−→ Φ(forget X )( M /G ) . To see that this is well-defined, let
G, H ⊂ M universal and let u ∈ M define an H -fixed point of M /G . Then we have commutative diagrams(1.2.6) Φ( X )( E M /G ) = maps E M ( E M /G, X ) X G Φ( X )( E M /H ) = maps E M ( E M /H, X ) X H (– · u ) ∗ ev [1] ∼ = u. –ev [1] ∼ = and, for each Y ∈ M -SSet ,(1.2.7) Φ( Y )( M /G ) = maps M ( M /G, Y ) Y G Φ( Y )( M /H ) = maps M ( M /H, Y ) Y H . (– · u ) ∗ ev [1] ∼ = u. –ev [1] ∼ = Taking Y = forget X in (1 . . τ X isnatural (and hence defines a morphism in Fun ( O op M , SSet )). It is then obvious that τ is natural (say, in the unenriched sense), as it is levelwise given by a compositionof natural transformations. △ We can now easily deduce the theorem: the vertical functors in (1 . .
5) in-duce equivalences on quasi-localizations by Proposition 1.1.36, and by the previouscorollary the bottom horizontal arrow is fully faithful with essential image thosepresheaves that invert centralizing isomorphisms. It follows that forget ∞ is fullyfaithful with essential image those M -simplicial sets X such that Φ( X ) invertscentralizing isomorphisms. But taking Y = X and a u ∈ M centralizing G = H in (1 . .
7) we see that Φ( X ) inverts centralizing morphisms if and only if X issemistable. This finishes the proof. (cid:3) G -GLOBAL HOMOTOPY THEORY G -global model structures. Let us fix some (possibly infinite) dis-crete group G . We now want to extend the above discussion to yield a G -globalmodel structure on the category E M - G -SSet of simplicial sets with a G -actionand a commuting E M -action, which we can equivalently think of as simplicial setswith an action of E M × G , or as the category of G -objects in E M -SSet .For G = 1 this will recover the previous model; however, as soon as G con-tains torsion, the weak equivalences wil be strictly finer than the underlying globalweak equivalences. In particular, we will show later in Theorem 1.2.83 that theweak equivalences are fine enough to recover proper G -equivariant homotopy theory ,i.e. the homotopy theory of G -simplicial sets with respect to those maps inducingweak equivalences on H -fixed points for finite subgroups H . Definition . A graph subgroup Γ = Γ
H,ϕ of M × G is called universal ifthe corresponding subgroup H ⊂ M is universal. A graph subgroup Γ ⊂ ( E M× G ) is universal if it is universal as a subgroup of M × G . Construction . Let H ⊂ M be a subgroup and let ϕ : H → G be agroup homomorphism. We write M × ϕ G := ( M × G ) / Γ H,ϕ and E M × ϕ G := ( E M × G ) / Γ H,ϕ . It follows immediately from the constructions that
M × ϕ G corepresents (–) ϕ on M - G -SSet and that E M × ϕ G corepresents (–) ϕ on E M - G -SSet . Corollary . There exists a unique model structure on the category M - G -SSet of M - G -simplicial sets in which a morphism f is a • weak equivalence if and only if f ϕ is a weak homotopy equivalence for anyuniversal H ⊂ M and any group homomorphism ϕ : H → G , • fibration if and only f ϕ is a Kan fibration for any universal H ⊂ M andany group homomorphism ϕ : H → G .This model structure is simplicial, combinatorial, proper, and filtered colimits in itare homotopical. A possible set of generating cofibrations is given by { ( M × ϕ G ) × ( ∂ ∆ n ֒ → ∆ n ) : n ≥ , H ⊂ M universal , ϕ : H → G homomorphism } and a possible set of generating acyclic cofibrations by { ( M × ϕ G ) × (Λ nk ֒ → ∆ n ) : 0 ≤ k ≤ n, H ⊂ M universal , ϕ : H → G } . We call it the G -universal model structure and its weak equivalences the G -universalweak equivalences . (cid:3) Corollary . There is a unique model structure on E M - G -SSet inwhich a map f is a weak equivalence or fibration if and only if it is so in M - G -SSet .This model structure is simplicial, combinatorial, proper, and filtered colimitsin it are homotopical. A possible set of generating cofibrations is given by { ( E M× ϕ G ) × ( ∂ ∆ n ֒ → ∆ n ) : n ≥ , H ⊂ M universal , ϕ : H → G homomorphism } and a possible set of generating acyclic cofibrations by { ( E M × ϕ G ) × (Λ nk ֒ → ∆ n ) : 0 ≤ k ≤ n, H ⊂ M universal , ϕ : H → G } We call this the G -global model structure and its weak equivalences the G -globalweak equivalences . (cid:3) .2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 41 Analogously to the global situation, we will now show that the forgetful functor E M - G -SSet → M - G -SSet is fully faithful on associated quasi-categories andcharacterize its essential image. Definition . An M - G -simplicial set X is called G -semistable if the( G × H )-equivariant map u. – : X → X is a G H,G -weak equivalence for any universal H ⊂ M and any u ∈ M centralizing H , i.e. for any group homomorphism ϕ : H → G the induced map u. – : X ϕ → X ϕ is a weak homotopy equivalence.1.2.3.1. Combinatorics of the G -global orbit category. As before, our compari-son will proceed indirectly via the respective orbit categories. The mapping spacesin this are again given by fixed points of quotients, and this section is devoted toclarifying their structure. In fact we will do all of this in greater generality now aswe will need the additional flexibility later. For this let us fix a group K togetherwith a (finitely or infinitely) countable faithful K -set A and a G - K -biset X . Remark . The set Inj(
A, ω ) of injections A → ω has a natural right K -action via u.k = u ( k. –); this obviously commutes with the left M -action comingfrom postcomposition, making Inj( A, ω ) into an M - K -biset. (If A = ω with itstautological K -action, then the above right action is just composition in M and wewill accordingly drop the dot denoting the action.)The right K -action is free for any A as above: if k = 1, then we find a ∈ A with k.a = a by faithfulness, and then ( u.k )( a ) = u ( k.a ) = u ( a ) by injectivity.We begin with the following trivial observation: Lemma . Let ( u , . . . , u n ; x ) , ( v , . . . , v n ; y ) ∈ Inj(
A, ω ) n +1 × X . (1) These represent the same n -simplex of E Inj(
A, ω ) × K X (where ‘ × K ’means that we divide out the diagonal right K -action) if and only if thereexists some k ∈ K such that v i = u i .k for all i = 0 , . . . , n and y = x.k . (2) Assume these indeed represent the same n -simplex and that there is some i such that u i = v i . Then x = y and u j = v j for all j = 0 , . . . , n . (3) Assume x = y . Then these represent the same n -simplex if and only ifthere is a k ∈ Stab K ( x ) (where Stab K denotes the stabilizer) such that v i = u i .k for i = 0 , . . . , n . Proof.
The first statement holds by definition and the third one is obviously aspecial case of this. Finally, the second statement follows from the first by freenessof the right K -action on Inj( A, ω ). (cid:3) We can now characterize the ϕ -fixed points for any universal H ⊂ M and anyhomomorphism ϕ : H → G , generalizing Lemma 1.2.22: Lemma . Let ( u , . . . , u n ; x ) ∈ Inj(
A, ω ) × X such that [ u , . . . , u n ; x ] ∈ ( E Inj(
A, ω ) × K X ) ϕ . Then there exists for each h ∈ H a unique σ ( h ) ∈ K suchthat (1.2.8) hu i = u i .σ ( h ) for all i = 0 , . . . , n .We have (1.2.9) x.σ ( h ) = ϕ ( h ) .x for all h ∈ H .The converse holds: whenever there exists a set map σ : H → K satisfying (1 . . and (1 . . , then [ u , . . . , u n ; x ] is a ϕ -fixed point. Moreover, σ is automat-ically a group homomorphism. G -GLOBAL HOMOTOPY THEORY Proof.
That ( u , . . . , u n ; x ) is ϕ -fixed means by definition that for each h ∈ H ( u , . . . , u n ; x ) ∼ ( h, ϕ ( h )) . ( u , . . . , u n ; x ) = ( hu , . . . , hu n ; ϕ ( h ) .x )which again means by definition that there exists a σ ( h ) ∈ K such that hu i = u i .σ ( h ) and moreover ϕ ( h ) .x = x.σ ( h ). This σ ( h ) is already uniquely characterizedby the first property (for i = 0) as K acts freely from the right on Inj( A, ω ), provingthe first half of the proposition.Conversely, if such a σ exists, then [ u , . . . , u n ; x ] is clearly ϕ -fixed. Moreover, u .σ ( hh ′ ) = hh ′ u = hu .σ ( h ′ ) = u .σ ( h ) σ ( h ′ ) , and hence σ ( hh ′ ) = σ ( h ) σ ( h ′ ) by freeness of the right K -action. (cid:3) In the situation of Lemma 1.2.34 we write σ ( u ,...,u n ) for the unique (homo-morphism) H → K satisfying (1 . . Corollary . Let σ : H → K be any group homomorphism. Then thereexists u ∈ Inj(
A, ω ) such that hu = u.σ ( h ) for all h ∈ H . Moreover, if x ∈ X satisfies (1 . . , then [ u ; x ] is a ϕ -fixed point of E Inj(
A, ω ) × K X . Proof.
When equipped with the tautological H -action, ω is a complete H -setuniverse; on the other hand, σ ∗ A is a countable H -set by assumption, so that thereexists an H -equivariant injection u : σ ∗ A → ω . The H -equivariance of u directlytranslates to hu = u.σ ( h ), and Lemma 1.2.34 then proves that [ u ; x ] is ϕ -fixed. (cid:3) The comparison.
Let us describe the orbit category O E M× G (withrespect to the universal graph subgroups): Remark . The objects of O E M× G are the E M - G -simplicial sets E M× ϕ G where H ⊂ M is universal and ϕ : H → G is a homomorphism. If K ⊂ M isanother universal subgroup and ψ : K → G a homomorphism, then Lemma 1.2.34tells us that any n -cell of maps O E M× G ( E M × ϕ G, E
M × ψ G ) can be representedby a tuple ( u , . . . , u n ; g ) ∈ M n +1 × G such that there exists a (necessarily unique)group homomorphism σ : H → K satisfying(1.2.10) hu i = u i σ ( h ) and ϕ ( h ) g = gψ ( σ ( h ))for all i = 0 , . . . , n and h ∈ H . Another such tuple ( u ′ , . . . , u ′ n ; g ′ ) representsthe same morphism if and only if there exists k ∈ K such that u ′ i = u i k for all i = 0 , . . . , n and g ′ = gψ ( k ).If L ⊂ M is another universal subgroup, θ : L → G a group homomorphismand if ( u ′ , . . . , u ′ n ; g ′ ) represents a morphism E M × ψ G → E M × θ G then thecomposition ( u ′ , . . . , u ′ n ; g ′ )( u , . . . , u n ; g ) is represented by ( u u ′ , . . . , u n u ′ n ; gg ′ ).Similarly, objects of O M× G are the M - G -sets M × ϕ G with ϕ as above andmaps can be represented by pairs ( u ; g ) with u ∈ M and g ∈ G satisfying analogousconditions to the above. Again, compositions are given by multiplication in M and G . In particular, we have a canonical functor i : O M× G → O E M× G sending M× ϕ G to E M × ϕ G and a morphism M × ϕ G → M × ψ G represented by ( u ; g ) to themorphism E M × ϕ G → E M × ψ G represented by the same pair ( u ; g ). Definition . A morphism f : M × ϕ G → M × ϕ G in O M× G is called G -centralizing if there exists a u ∈ M centralizing H such that f is represented by( u ; 1). Analogously, we define G -centralizing morphisms in O E M× G . .2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 43 Proposition . The map i : O M× G → O E M× G is a simplicial localiza-tion at the G -centralizing morphisms. Proof.
Let us write W ⊂ ( O E M× G ) for the subcategory of G -centralizingmorphisms. As in the non-equivariant setting (Proposition 1.2.25) it is enough toprove that for each n ≥ s ∗ : (( O E M× G ) , W ) → (( O E M× G ) n , s ∗ W )induced by the unique map s : [ n ] → [0] is a homotopy equivalence. As before wehave a strict left inverse given by restriction along i : [0] → [ n ] , s ∗ i ∗ and the identity.For this we recall from the proof of Proposition 1.2.25 that we can choose foreach universal subgroup H ⊂ M injections α H , β H ∈ M centralizing H and suchthat ω = im( α H ) ∐ im( β H ).Now let H, K ⊂ M be universal and let ϕ : H → G and ψ : K → G be grouphomomorphisms. Then any morphism E M× ϕ G → E M× ψ G in ( O E M× G ) n can berepresented by a tuple ( u , . . . , u n ; g ) such that there exists a group homomorphism σ : H → K satisfying the relations (1 . . i = 0 , . . . , n a unique v i such that v i α K = α H u i and v i β K = β H u . We claim that ( v , . . . , v n ; g ) again defines a morphism, i.e. its class in E M × ψ G is ϕ -fixed. Indeed, we have seen in the non-equivariant case that hv i = v i σ ( h ), hence( h, . . . , h ; ϕ ( h ))( v , . . . , v n ; g ) = ( hv , . . . , hv n ; ϕ ( h ) g )= ( v σ ( h ) , . . . , v n σ ( h ); gψ ( σ ( h ))) ∼ ( v , . . . , v n ; g )as desired. Similarly, one uses the argument from the non-equivariant case to showthat the morphism represented by ( v , . . . , v n ; g ) does not depend on the chosenrepresentative ( u , . . . , u n ; g ).We now define a functor f : ( O E M× G ) n → ( O E M× G ) n as follows: f is theidentity on objects and on morphisms given by the above construction. Using thatthe above is independent of choices one easily checks that f is indeed a functor. Asbefore we have by construction natural transformations id ⇐ f ⇒ s ∗ i ∗ , where theleft hand transformation is given on E M × ϕ G by ( α H , . . . , α H ; 1) and the righthand one by ( β H , . . . , β H ; 1). As α H and β H centralize H by definition, these areweak equivalences, finishing the proof. (cid:3) By the same arguments as in the non-equivariant setting we can deduce:
Theorem . The adjunction E M × L M – : M - G -SSet ∞ G -universal ⇄ E M - G -SSet ∞ : forget ∞ is a Bousfield localization; in particular, forget ∞ is fully faithful. Moreover, itsessential image consists precisely of the G -semistable M - G -simplicial sets. (cid:3) Additional model structures.
The following corollary lifts the abovecomparison of quasi-categories to the level of model categories:
Corollary . There is a (unique) model structure on M - G -SSet inwhich a map f : X → Y is • a weak equivalence iff E M× L M f is an isomorphism in Ho( E M - G -SSet ) , • a cofibration iff it is so in the usual model structure on M - G -SSet . G -GLOBAL HOMOTOPY THEORY An object X ∈ M - G -SSet is fibrant in this model structure if and only if it isfibrant in the usual model structure and moreover G -semistable in the sense of Def-inition 1.2.31. We call this the G -global model structure and its weak equivalencesthe G -global weak equivalences .This model structure is combinatorial with generating cofibrations { ( M × ϕ G ) × ( ∂ ∆ n ֒ → ∆ n ) : n ≥ , H ⊂ M universal , ϕ : H → G homomorphism } , simplicial (with respect to the obvious simplicial enrichment), left proper, and more-over filtered colimits in it are homotopical. Finally, the adjunction E M × M – : M - G -SSet G -global ⇄ E M - G -SSet : forget is a Quillen equivalence with homotopical right adjoint. Proof.
Theorem 1.2.39 allows us to invoke Lurie’s localization criterion (The-orem A.2.5) which proves all of the above claims except for the part about filteredcolimits, which is in turn an instance of Lemma A.2.6. (cid:3)
As a special case of Corollary 1.1.39, the G -global weak equivalences of E M - G -simplicial sets are part of an injective model structure. We will now prove theanalogue of this for M - G -SSet : Theorem . There exists a unique model structure on M - G -SSet whosecofibrations are the underlying cofibrations and whose weak equivalences are the G -global weak equivalences.We call this the injective G -global model structure . It is combinatorial, sim-plicial, left proper, and filtered colimits in it are homotopical.Finally, the simplicial adjunction forget: E M - G -SSet inj. G -global ⇄ M - G -SSet inj. G -global : maps M ( E M , –) is a Quillen equivalence with homotopical left adjoint. Proof.
We first claim that the G -global weak equivalences are stable underpushout along injective cofibrations. For this we let f : A → B be a G -global weakequivalence and i : A → C an injective cofibration. Applying the factorizationaxiom in the G -global model structure, we can factor f = pk where k is an acycliccofibration and p is an acyclic fibration. As the G -global and universal modelstructures on M - G -SSet have the same cofibrations, they also have the sameacyclic fibrations; in particular, p is a universal weak equivalence.We now consider the iterated pushout A X BC Y D. fki pjℓ q
Then ℓ is an acyclic cofibration in the G -global model structure as a pushout ofan acyclic cofibration. Moreover, j is an injective cofibration as a pushout of aninjective cofibration, so q is a G -universal (and hence in particular G -global) weakequivalence by left properness of the equivariant injective model structure. Theclaim follows as qℓ is a pushout of f along i . .2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 45 We therefore conclude from Corollary A.2.11 that the model structure exists,that it is combinatorial and left proper. Moreover, Lemma A.2.6 shows that filteredcolimits in it are still homotopical, so it only remains to verify the pushout productaxiom for the simplicial tensoring.For this we can argue precisely as in the proof of Corollary 1.1.39 once we showthat for any simplicial set K the functor K × – preserves G -global weak equivalences,and that for any M - G -simplicial set X the functor – × X sends weak equivalencesof simplicial sets to G -global weak equivalences.For the second statement it is actually clear that – × X sends weak equivalenceseven to G -universal weak equivalences. For the first statement it is similarly clearthat K × – preserves G -universal weak equivalences, but it also preserves acycliccofibrations in the G -global model structure as the latter is simplicial. The claimagain follows as any G -global weak equivalence can be factored as a G -global acycliccofibration followed by a G -universal weak equivalence. (cid:3) G -semistable replacement. If one wants to check if amorphism f : X → Y in G - M -SSet is a G -global weak equivalence straight fromthe definition, one runs into trouble as soon as at least one of X or Y is not cofibrantbecause we then have to take cofibrant replacements first, and the ones providedby the small object argument are completely intractable.On the other hand, the G -universal weak equivalences are much easier to check,so one could instead try to take G -semistable replacements of X and Y and thenapply the following easy observation: Lemma . A morphism f : X → Y of G -semistable M - G -simplicial sets isa G -global weak equivalence if and only if it is a G -universal weak equivalence. (cid:3) However, this leaves us with the problem of finding (functorial) G -semistablereplacements. While the G -global model structure asserts that these exist, theyare again completely inexplicit. We could also try to construct these by means of E M × L M –, but then we are of course back where we started.In this subsection we will remedy this situation by constructing explicit G -semistable replacements, yielding a characterization of the G -global weak equiva-lences. This characterization will become crucial later (see e.g. Theorem 1.4.22)and in particular I do not know how to prove the respective statements ‘by hand.’ Remark . Before we come to the construction, let us think about howit should look like. The simplicial set E M is (weakly) contractible, and one canconclude from this, see e.g. [ Lur09 , Proposition 4.2.4.4], that E M -SSet mod-els ordinary non-equivariant homotopy theory when equipped with the underlying weak equivalences. More precisely, with respect to these weak equivalences, thehomotopical functorsconst : SSet → E M -SSet and forget: E M -SSet → SSet induce mutually quasi-inverse equivalences on associated quasi-categories. It followsthat the composition M -SSet ∞ G -universal E M× L M – −−−−−−→ E M -SSet ∞ forget −−−→ SSet ∞ agrees with taking homotopy colimits over M . G -GLOBAL HOMOTOPY THEORY Action categories.
The remark suggests that we might be lucky andsucceed in constructing the replacement by means of a suitable equivariant en-hancement of one of the standard constructions of homotopy colimits. This willindeed work for the model of what is usually called the action groupoid (though itwon’t be a groupoid in our case).
Construction . Let X be any M -set. We write X // M for the ‘actioncategory,’ i.e. the category with set of objects X and for any x ∈ X and u ∈ M amorphism u : x → u.x ; we emphasize that this means that if u = v with u.x = v.x then u and v define two distinct morphisms x → u.x = v.x . The composition in X // M is given by multiplication in M .The M -action on X immediately gives an M -action on Ob( X // M ); however,it is not entirely clear how to extend this to morphisms. For an invertible element α ∈ core M , the condition that α.f for f : x → y should be a morphism α.x → α.y naturally leads to the guess α.f = αf α − . While general elements of M are notinvertible, there is still a notion of conjugation . This is made precise by the followingconstruction, which is implicit in [ Sch08 , proof of Lemma 5.2] (which Schwedeattributes to Strickland) and also appeared in an earlier version of [
Sch19b ]: Construction . Let α ∈ M . We define for any u ∈ M the ‘conjugation’ c α ( u ) of u by α via c α ( u )( x ) = ( αu ( y ) if x = α ( y ) x if x / ∈ im α. We remark that this is well-defined (as α is injective) and one easily checks thatthis is again injective, so that we get a map c α : M → M .Put differently, c α ( u ) is the unique element of M such that c α ( u ) α = αu and c α ( u )( x ) = x for x / ∈ im α. The first condition justifies the name ‘conjugation,’ and it means in particular that c α ( u ) = αuα − for invertible α .For a group G , conjugation by a fixed element g defines an endomorphism of G , and letting g vary this yields an action of G on itself. The analogous statementholds for the above construction: Lemma . For any α ∈ M , the map c α : M → M is a monoid homomor-phism. Moreover, for varying α this defines an action of M on itself, i.e. c = id M and c α ◦ c β = c αβ for all α, β ∈ M . Proof.
We will only prove the first statement, the calculations for the otherclaims being similar. For this let u, v ∈ M . Then c α ( uv ) α = αuv = c α ( u ) αv = c α ( u ) c α ( v ) α. On the other hand, if x / ∈ im( α ), then c α ( uv )( x ) = x and c α ( v )( x ) = x / ∈ im α ,hence also ( c α ( u ) c α ( v ))( x ) = x . We conclude that c α ( uv ) = c α ( u ) c α ( v ). Moreover,clearly c α (1) = 1, so that c α is indeed a monoid homomorphism. (cid:3) Construction . We define a M -action on X // M as follows: the actionof M on objects is the one on X , and on morphisms α ∈ M acts by sending x u −→ u.x to α.x c α ( u ) −−−→ α. ( u.x ); .2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 47 note that this is indeed a morphism as c α ( u ) . ( α.x ) = ( c α ( u ) α ) .x = ( αu ) .x = α. ( u.x )by construction of c α . As c α is a monoid homomorphism, the above indeed definesan endofunctor of X // M ; the previous lemma then implies that this is an M -action.If f : X → Y is a map of M -sets, then we write f // M for the functor that isgiven on objects by f and that sends a morphism x u −→ u.x to f ( x ) u −→ u.f ( x ) = f ( u.x ) . One easily checks that this is welldefined, functorial in f , and that f // M is M -equivariant. Postcomposing with the nerve we therefore get a functor M -Set → M -SSet that we denote by (–)// M again. If G is any group, we moreover get aninduced functor M - G -Set → M - G -SSet by pulling through the G -action (againdenoted by the same symbol).If X is any M -set, then there is a unique functor from X (viewed as a discretecategory) to X // M that is the identity on objects. This then yields a naturaltransformation π : discr ⇒ (–)// M from the functor that sends an M - G -set X tothe discrete simplicial set X with the induced action. Construction . Let X be any M - G -simplicial set. Applying the aboveconstruction levelwise yields a bisimplicial set X M with ( X M ) n, • = X n // M and this receives a map from the bisimplicial set Discr X with (Discr X ) n, • =discr X n . This yields a functor M - G -SSet → M - G -BiSSet receiving a naturaltransformation Π : Discr ⇒ M - G -SSet → M - G -SSet , thatwe denote again by (–)// M , together with a natural transformation id ⇒ (–)// M ,that we again denote by π . We remark that on M - G -sets (viewed as discretesimplicial sets) this recovers the previous construction.1.2.4.2. The detection result.
Now we are ready to state the main result of thissubsection:
Theorem . (1) For any X ∈ M - G -SSet , the map π X : X → X // M is a G -global weak equivalence and X // M is G -semistable. (2) For any f : X → Y in M - G -SSet the following are equivalent: (a) f is a G -global weak equivalence. (b) f // M is a G -global weak equivalence. (c) f // M is a G -universal weak equivalence. The proof of this will occupy the rest of this subsection.
Remark . We defined the bisimplicial set X M in terms of the actioncategory construction. We can also look at this bisimplicial set ‘from the otherside,’ which recovers the bar construction (just as in the usual construction of non-equivariant homotopy quotients):If Y is any M - G -set, then ( Y // M ) m consists by definition of the m -chains y u −→ u .y u −→ ( u u ) .y u −→ · · · u m −−→ ( u m · · · u ) .y of morphisms in Y // M . Such a chain is obviously uniquely described by the source y ∈ Y together with the injections u m , . . . , u , u ∈ M , which yields a bijection( Y // M ) m ∼ = M m × Y . This bijection becomes M - G -equivariant, when we let G act via its action on X and M via its action on X and the conjugation action oneach of the M -factors. G -GLOBAL HOMOTOPY THEORY The assignment Y
7→ M m × Y becomes a functor in Y in the obvious way, andwith respect to this the above bijection is clearly natural in M -equivariant maps.Applying this levelwise, we therefore get a natural isomorphism(1.2.11) ( X M ) • ,m ∼ = M m × X of M - G -simplicial sets. While we will not need this below, we remark that unrav-elling the definitions, one can work out that under the isomorphism (1 . .
11) thestructure maps correspond to either multiplication in M or the action on X , sothis is indeed the bar construction as claimed above.By construction and the previous remark, we understand the bisimplicial set X M in both its simplicial directions individually. In non-equivariant simplicialhomotopy theory, the Diagonal Lemma then often allows to leverage this to provestatements about the diagonal ( X // M in our case). Luckily, this immediatelycarries over to our situation: Lemma . Let M be a monoid, let F be a collection of subgroups of M ,and let f : X → Y be a map of M -bisimplicial sets. Assume that for each n ≥ the map f n, • : X n, • → Y n, • is a F -weak equivalence, or that for each n ≥ the map f • ,n : X • ,n → Y • ,n is. Then also diag f : diag X → diag Y is a F -weak equivalence. Proof.
By symmetry it suffices to consider the first case. If H ∈ F is any sub-group, then ( f n, • ) H agrees literally with ( f H ) n, • (if we take the usual constructionof fixed points), and likewise (diag f ) H = diag( f H ). Therefore, the claim followsimmediately from the usual Diagonal Lemma, see e.g. [ GJ99 , Theorem 4.1.9]. (cid:3)
Let us draw some non-trivial consequences from this:
Corollary . The above functor (–)// M preserves G -universal weakequivalences of M - G -simplicial sets. Proof.
By the previous lemma (applied to the universal graph subgroups ofthe monoid
M × G ), it suffices that each (– M ) • ,n does, which follows from theisomorphism (1 . .
11) together with Lemma 1.1.2. (cid:3)
Lemma . Let X be any M - G -simplicial set. Then X // M is G -semistable. Proof.
Let H ⊂ M be any subgroup (in fact H need not be universal oreven finite for the argument below) and let α ∈ M centralize H . We want to showthat the ( H × G )-equivariant map α. – : X // M → X // M is a weak equivalencewith respect to the family of graph subgroups and we will show that it is even an( H × G )-equivariant homotopy equivalence.By Lemma 1.2.51 we reduce this to the case that X is a M - G -set , in whichcase X // M is the nerve of the ‘action category.’ We then observe that the maps α : x → α.x assemble into a natural transformation a : id ⇒ ( α. –) by virtue of theidentity c α ( u ) α = αu . This natural transformation is obviously G -equivariant, butit is also H -equivariant: if h ∈ H and x ∈ X are arbitrary, then h.a x is by definitionthe map hαh − : h.x → h. ( α.x ); as h commutes with α , this agrees with a h.x asdesired. Upon taking nerves, a therefore induces a ( H × G )-equvariant homotopybetween the identity and α. – as desired. (cid:3) The following two statements are again easily deduced from the isomorphism(1 . .
11) and we omit their proofs. .2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 49 Corollary . The functor (–)// M : G - M -SSet → G - M -SSet is co-continuous and it preserves injective cofibrations. (cid:3) Corollary . The functor (–)// M : G - M -SSet → G - M -SSet has anatural simplicial enrichment and with respect to this it preserves tensors. Thenatural transformation π is simplicially enriched. (cid:3) The only remaining input that we need to prove the theorem is an explicitcomputation of (–)// M on the M - G -sets M× ϕ G for H ⊂ M universal and ϕ : H → G any group homomorphism. We will actually prove the following much moregeneral statement, and this additional generality will become important later inthe proof of Theorem 1.4.22: Theorem . Let H ⊂ M be universal, let A be a countable faithful H -set, and let X be any G - H -biset. If we consider Inj(
A, ω ) as an M - H -biset inthe obvious way, then the unique functor Inj(
A, ω )//
M → E Inj(
A, ω ) that is theidentity on objects induces a G -universal weak equivalence (1.2.12) (cid:0) Inj(
A, ω ) × H X (cid:1) // M ∼ = (cid:0) Inj(
A, ω )// M (cid:1) × H X ∼ −→ (cid:0) E Inj(
A, ω ) (cid:1) × H X. Here the unlabelled isomorphism comes from the cocontinuity of (–)// M . Remark . The theorem also holds for A uncountable (as then both sidesare just empty). However, some lemmas we will appeal to only hold for countable A and this is also the only case we will be interested in. Therefore, we have decidedto state the theorem in the above form.The proof of the theorem needs some preparations and will be given at the endof this subsection. Let us already use it to deduce the detection result: Proof of Theorem 1.2.49.
We will only prove the first statement; the sec-ond one will then follow formally from this (cf. Lemma 1.2.42.)We already know by Lemma 1.2.53 that X // M is G -semistable for any X ∈ G - M -SSet . It therefore only remains to prove that π X : X → X // M is a G -global weak equivalence. As we have seen in Corollary 1.2.52 that (–)// M pre-serves G -universal weak equivalences, it suffices to prove this for cofibrant X , andCorollary 1.2.54 together with Corollary 1.1.8 reduces this further to the casethat X is the source or target of one of the standard generating cofibrations,i.e. X = ( M × ϕ G ) × ∂ ∆ n or X = ( M × ϕ G ) × ∂ ∆ n for some n ≥
0. By Corol-lary 1.2.55 we are then further reduced to X = M × ϕ G .In this case we observe that the unit η of the adjunction E M × M (–) ⊣ forget,evaluated at M × ϕ G can be factored as M × ϕ G π −→ ( M × ϕ G )// M ( . . ) −−−−−−→ ( E M ) × ϕ G ∼ = forget (cid:0) E M × M ( M × ϕ G ) (cid:1) , where the middle arrow comes from applying Theorem 1.2.56 for A = ω and X = G with its left regular G -action and H acting from the right via τ , and the finalisomorphism uses that the left adjoint E M × M – is cocontinuous. The abovecomposition is a G -global weak equivalence by Corollary 1.2.40, and the middlearrow is even a G -universal weak equivalence by Theorem 1.2.56. Therefore, 2-out-of-3 implies that π is a G -global weak equivalence as desired. By the abovereduction, this completes the proof. (cid:3) It remains to prove that the map (1 . .
12) is a G -universal weak equivalence. G -GLOBAL HOMOTOPY THEORY Example . As the proof of the theorem will be quite technical, let usbegin with something much easier that will nevertheless give an idea of the generalargument: we will show that the map in question is a non-equivariant weak equiva-lence for A finite and H = 1, which amounts to saying that Inj( A, ω )// M is weaklycontractible as a simplicial set. This argument also appears as [ SS20 , Example 3.3].By construction, the category Inj(
A, ω )// M has objects the injections i : A → ω and it has for every u ∈ M and i ∈ Inj(
A, ω ) a map u : i → ui . If now i, j ∈ Inj(
A, ω )are arbitrary, then we can pick a bijection u ∈ M with ui = j , i.e. i and j are iso-morphic in Inj( A, ω )// M . It follows that for our favourite i ∈ Inj(
A, ω ) the inducedfunctor B End( i ) → Inj(
A, ω )// M is an equivalence of categories. Therefore it suf-fices that the monoid End( i ) has weakly contractible classifying space. However,End( i ) consists precisely of those u ∈ M that fix im( i ) pointwise. Picking a bijec-tion ω ∼ = ω r im( i ) therefore yields an isomorphism of monoids M ∼ = End( i ). Since M has weakly contractible classifying space by [ Sch08 , Lemma 5.2] (whose proofSchwede attributes to Strickland), this finishes the proof.1.2.4.3.
Equivariant analysis of action categories.
Fix a universal subgroup K ⊂ M and a homomorphism ϕ : K → G . By definition, the simplicial set(Inj( A, ω ) × H X )// M is the nerve of the category of the same name, and as thenerve is a right adjoint, this is compatible with ϕ -fixed points. In the following, wewant to understand these fixed point categories better and in particular describethem as disjoint unions of monoids in analogy with the above example.However, in Theorem 1.2.56 we allow A to be infinite (and A = ω is the casewe actually used in the proof of Theorem 1.2.49). In this case, there are ‘toomany isomorphism classes’ in (Inj( A, ω )// M ) × H X : for example, not all objects in M // M are isomorphic, though they all receive a map from 1 ∈ M . To salvage thissituation we introduce the full subcategory C K ⊂ (Inj( A, ω ) × H X )// M spannedby those [ u, x ] for which im( u ) c ⊂ ω contains a complete K -set universe (this isindependent of the choice of representative, as im( u ) = im( u.h ) for all h ∈ H ). Lemma . The inclusion C K ֒ → (Inj( A, ω )// M ) × H X induces a ( K × G ) -homotopy equivalence on nerves. Proof.
Let α ∈ M be K -equivariant such that im( α ) c contains a complete K -set universe. Then α. – is ( K × G )-equivariant and it takes all of (Inj( A, ω )// M ) × H X to C K . We claim that this is a ( K × G )-homotopy inverse to the inclusion. Indeed,the proof of Lemma 1.2.53 shows that the maps α : x → α.x define a naturaltransformation from the identity to α. – as endofunctors of (Inj( A, ω )// M ) × H X ,showing that α. – is a right ( K × G )-homotopy inverse. However, as C K is a fullsubcategory, these also define such a transformation if we view both as endofunctorsof C K , proving that α. – is also a left ( K × G )-homotopy inverse. (cid:3) The next lemma in particularly tells us that C K avoids the aforementionedissue. For its proof we need the following notation: Definition . Let
A, B be sets, let A = A ⊔ A be any partition, andlet f i : A i → B ( i = 1 ,
2) be any maps of sets. Then we write f + f for the uniquemap A → B that agrees on A with f and on A with f .By slight abuse, we will also apply the above when f and f are maps intosubsets of B . Obviously, f + f will be injective if and only if f and f areinjections with disjoint image. .2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 51 Lemma . Let p, q ∈ C ϕK and fix a representative ( u, x ) ∈ Inj(
A, ω ) × X of p . Then the following are equivalent: (1) There exists a map f : p → q in C ϕK . (2) There exists a representative of q of the form ( v, x ) such that in addition σ u = σ v (see the discussion after Lemma 1.2.34). (3) There exists an isomorphism f ′ : p → q in C ϕK . Proof.
Obviously, (3) ⇒ (1). We will prove that also (1) ⇒ (2) and (2) ⇒ (3).Assume f : p → q is any morphism in C ϕK . Then q = f.p , so that ( f u, x ) is arepresentative of q . We claim that this has the desired property, i.e. v := f u satisfies σ v = σ u . But indeed, as f is ϕ -fixed, it has to be K -equivariant by definition ofthe action. Then v.σ v ( k ) = kv = kf u = f ku = f u.σ u ( k ) = v.σ u ( k )and hence σ v ( k ) = σ u ( k ) as H acts freely on Inj( A, ω ). This proves (1) ⇒ (2).For the proof of (2) ⇒ (3), we observe that im( u ) and im( v ) are both K -subsets of ω : indeed, k.u ( a ) = ( ku )( a ) = u ( σ u ( k ) .a ) for all a ∈ A , and similarlyfor v . Hence, ω r im( u ) and ω r im( v ) are K -sets in their own right; as theyare obviously countable and moreover contain a complete K -set universe each bydefinition of C , they are both themselves complete K -set universes. It follows thatthere exists a K -equivariant bijection f ′ : ω r im( u ) ∼ = ω r im( v ).On the other hand f ′ = vu − : im( u ) → im( v ) is also K -equivariant because f ′ ( k.u ( a )) = f ′ ( u ( σ u ( k ) .a )) = v ( σ u ( k ) .a ) = v ( σ v ( k ) .a ) = k.v ( x ) = k.f ′ ( u ( a ))for all k ∈ K , a ∈ A . As it is moreover obviously bijective, we conclude that f ′ := f ′ + f ′ defines an isomorphism [ u, x ] → [ v, x ] in C ϕK as desired. (cid:3) We fix for each isomorphism class of C ϕK a representative p ∈ C ϕK and for eachsuch p in turn a representative ( u, x ) ∈ Inj(
A, ω ) × X . Let us write I for the setof all these. The above lemma then implies: Corollary . The tautological functor
Φ : a ( u,x ) ∈ I B End C ϕK ([ u, x ]) → C ϕK is an equivalence of categories. (cid:3) Let us now study these monoids more closely:
Proposition . Let ( u, x ) ∈ Inj(
A, ω ) × X such that [ u, x ] ∈ C ϕK . (1) Let f : [ u, x ] → [ u, x ] be any map in C ϕK . Then there exists a unique τ ( f ) ∈ H such that f u = u.τ ( f ) . Moreover, τ ( f ) centralizes im σ u andstabilizes x ∈ X . (2) The assignment f τ ( f ) defines a monoid homomorphism τ : End C ϕK ([ u, x ]) → C H (im σ u ) ∩ Stab H ( x ) . (We caution the reader that τ depends on the chosen representative.) (3) The homomorphism τ induces a weak equivalence on classifying spaces. For the proof of the proposition we will need:
Lemma . The monoid M K of K -equivariant injections ω → ω has weaklycontractible classifying space. G -GLOBAL HOMOTOPY THEORY Proof.
This is an equivariant version of [
Sch08 , proof of Lemma 5.2]. As K is universal, we can pick a K -equivariant bijection ω ∼ = ω ∐ ω which yields two K -equivariant injections α, β ∈ M whose images partition ω . Then the conjugationhomomorphism c α : M → M satisfies c α ( u ) α = αu and c α ( u ) β = β for all u ∈ M . The first identity proves that α defines a natural transformation fromthe identity to B ( c α ) : B M → B M (also cf. the proof of Lemma 1.2.53) while thesecond one shows that β defines a natural transformation from the constant functorto it. Upon taking nerves, we therefore get a zig-zag of homotopies between theidentity and a constant map, proving the claim. (cid:3) Proof of Proposition 1.2.63.
For the first statement we observe that thereexists at most one such τ ( f ) by freeness of the action. On the other hand, f beinga morphism in particular means that f. [ u, x ] = [ u, x ]. Plugging in the definitionof the action and of the equivalence relation we divided out, this means that thereexists τ ( f ) ∈ H with ( f u, x ) = ( u.τ ( f ) , x.τ ( f )). Thus, it only remains that τ ( f )centralizes im( σ u ). Indeed, if k ∈ K is arbitrary, then on the one hand kf u = f ku = f u.σ u ( k ) = u. ( τ ( f ) σ u ( k ))(where we have used that f is K -equivariant since it is a morphism in C ϕK ) and onthe other hand kf u = ku.τ ( f ) = u. ( σ u ( k ) τ ( f )) . Thus, u. ( τ ( f ) σ u ( k )) = u. ( σ u ( k ) τ ( f )), whence indeed τ ( f ) σ u ( k ) = σ u ( k ) τ ( f ) bythe freeness of the right H -action. This finishes the proof of (1).For the second statement, we observe that f f ′ u = f u.τ ( f ′ ) = u. ( τ ( f ) τ ( f ′ ))and hence τ ( f f ′ ) = τ ( f ) τ ( f ′ ) by the above uniqueness statement. Analogously,1 u = u = u. τ (1) = 1.For the final statement, we will first prove: Claim.
The assignmentT : End C ϕK ([ u, x ]) → Inj( ω r im u, ω r im u ) K × (cid:0) C H (im σ u ) ∩ Stab H ( x ) (cid:1) f (cid:0) f | ω r im( u ) : ω r im( u ) → ω r im( u ) , τ ( f ) (cid:1) defines an isomorphism of monoids. Proof.
This is well-defined by the first part and since the injection f restrictsto a self-bijection of im( u ) by the above, so that it also has to preserve ω r im( u ).It is then obvious (using the second part of the proposition) that T is a monoidhomomorphism.We now claim that it is actually an isomorphism of monoids. For injectivity, itsuffices to observe that if τ ( f ) = τ ( f ′ ), then f u = uτ ( f ) = uτ ( f ′ ) = f ′ u .For surjectivity, we let f : ω r im( u ) → ω r im( u ) be any K -equivariant injectionand we let t ∈ C H (im σ u ) ∩ Stab H ( x ). Then there is a unique map f : im( u ) → im( u ) with f u = u.t and this is automatically injective (in fact, even bijective).We claim that it is also K -equivariant. Indeed, if k ∈ K is arbitrary, then k.f ( u ( x )) = ku ( t.x ) = u ( σ u ( k ) t.x ) = u ( tσ u ( k ) .x ) = f ( u ( σ u ( k ) .x )) = f ( k.u ( x ))where we have used that t commutes with σ u ( k ). Hence, f := f + f definesa K -equivariant injection ω → ω and if we can show that this is a morphism .2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 53 [ u, x ] → [ u, x ] in C K , then this will be the desired preimage of ( f , t ). Indeed, f. [ u, x ] is represented by( f u, x ) = ( u.t, x ) ∼ ( u, x.t − ) = ( u, x ) , where we have used that t and hence also t − stabilizes x . △ To prove that τ induces a weak homotopy equivalence on classifying spaces wenow observe that the induced map factors asN (cid:0) B End([ u, x ]) (cid:1) N( B T) −−−−→ N (cid:0) B (Inj( ω r im u, ω r im u ) K × L ) (cid:1) ∼ = N (cid:0) B (Inj( ω r im u, ω r im u ) K ) (cid:1) × N( BL ) pr −→ N( BL ) , where L := C H (im σ u ) ∩ Stab H ( x ). The first map is an isomorphism by the previousstatement, so it suffices that Inj( ω r im u, ω r im u ) K has trivial classifying space.But indeed, as in the proof of Lemma 1.2.61 we see that ω r im u is a complete K -set universe, so that Inj( ω r im u, ω r im u ) K ∼ = M K as monoids. Thus, theclaim follows from Lemma 1.2.64. (cid:3) Equivariant analysis of quotient categories.
We recall that the simpli-cial set E Inj(
A, ω ) × X is canonically isomorphic to the nerve of the groupoid ofthe same name. On the other hand, the right H -action on E Inj(
A, ω ) × X is free(by faithfulness of A ), and it is well-known that the nerve preserves quotients by free group actions, so we conclude that the simplicial set E Inj(
A, ω ) × H X is againcanonically identified with the nerve of the corresponding quotient of categories ,which we denote by the same name.In the following we want to devise a description of this category and its fixedpoints analogous to the above results. For this we first observe that while hom setsin quotient categories are in general hard to describe, the situation is easier herebecause this particular quotient is preserved of the nerve: namely, any morphism p → q (for p, q ∈ Inj(
A, ω ) × K X ) is represented by a triple ( u, v ; x ) with u, v ∈ Inj(
A, ω ), x ∈ X such that [ u ; x ] = q and [ v, x ] = p ; moreover, a triple ( u ′ , v ′ ; x ′ )represents the same morphism if and only if there exists an h ∈ H with u ′ = u.h , v ′ = v.h , x ′ = x.h . The following lemma gives a more concrete description once wehave fixed representatives of p and q : Lemma . Let ( u, x ) , ( v, y ) ∈ Inj(
A, ω ) × X . Then we have a bijection { h ∈ H : y.h = x } → Hom E Inj(
A,ω ) × H X ([ u ; x ] , [ v ; y ]) h [ v.h, u ; x ] = [ v, u.h − , y ] . In particular, the assignment (1.2.13) Stab H ( x ) → End E Inj(
A,ω ) × H X ([ u ; x ]) h [ u.h, u ; x ] is bijective; in fact, this even defines an isomorphism of groups. Proof.
Let us denote the above map by α . We first observe that for any h on the left hand side the representative ( v.h, u ; x ) differs from ( v, u.h − , y ) only byright multiplication by h , so that the two given definitions of α ( h ) indeed agree. Inparticular, they define an edge from [ u ; x ] to [ v ; y ], proving that α is well-defined.Lemma 1.2.33-(2) immediately implies that α is injective. To see that α issurjective, we pick an edge on the right hand side and let ( a, b ; c ) be a representative. G -GLOBAL HOMOTOPY THEORY By definition [ b ; c ] = [ u ; x ], so after acting suitably on the right by H on ( a, b ; c ), wemay assume b = u and c = x , i.e. our chosen representative takes the form ( a, u ; x ).But by assumption this represents an edge to [ v ; y ] and hence [ a ; x ] = [ v ; y ], i.e. thereexists an h ∈ H such that a = v.h and x = y.h , which then obviously defines thedesired preimage.Specializing to ( v ; y ) = ( u ; x ) shows that (1 . .
13) is bijective. The calculation[ u.hh ′ , u ; x ] = [ u.hh ′ , u.h ′ ; x ][ u.h ′ , u ; x ] = [ u.h, u ; x ][ u.h ′ , u ; x ](where the final equality uses that x. ( h ′ ) − = x ) then shows that it is in fact anisomorphism of groups. (cid:3) Similarly, we can describe the hom sets in the fixed point categories:
Lemma . Let ( u, x ) , ( v, y ) ∈ Inj(
A, ω ) × X such that [ u ; x ] , [ v ; y ] are ϕ -fixed points of E Inj(
A, ω ) × H X . Then { h ∈ H : y.h = x, hσ u ( k ) h − = σ v ( k ) ∀ k ∈ K } → Hom ( E Inj(
A,ω ) × H X ) ϕ ([ u ; x ] , [ v ; y ]) h [ v.h, u ; x ] = [ v, u.h − , y ] . is well-defined and bijective. In particular, this yields a bijection Stab H ( x ) ∩ C H (im σ u ) → End ( E Inj(
A,ω ) × H X ) ϕ ([ u ; x ]) h [ u.h, u ; x ] . This map is in fact even an isomorphism of groups.
Proof.
By the previous lemma we are reduced to proving that [ v.h, u ; x ] is ϕ -fixed if and only if σ v ( k ) = hσ u ( k ) h − for all k ∈ K . Indeed,( k, ϕ ( k )) . ( v.h, u ; x ) = ( kv.h, k.u ; ϕ ( k ) .x ) = ( v.σ v ( k ) h, u.σ u ( k ); x.σ u ( k )) ∼ (cid:0) v. ( σ v ( k ) hσ u ( k ) − ) , u ; x (cid:1) where we have applied Lemma 1.2.34 to ( u ; x ). By freeness of the right H -actionthis represents the same element as ( v.h, u ; x ) if and only if σ v ( k ) hσ u ( k ) − = h ,which is obviously equivalent to the above condition. (cid:3) Proposition . The functor
Ψ : a ( u,x ) ∈ I B (cid:0) C H (im σ u ) ∩ Stab H ( x ) (cid:1) → (cid:0) E Inj(
A, ω ) × H X (cid:1) ϕ given on the ( u, x ) -summand by sending t ∈ C H (im σ u ) ∩ Stab H ( x ) to the morphism [ u.t, u ; x ] is an equivalence of groupoids. Proof.
Lemma 1.2.66 implies that this indeed lands in the ϕ -fixed points andthat each of the mapsC H (im σ u ) ∩ Stab H ( x ) → End (cid:0) E Inj(
A,ω ) × H X (cid:1) ϕ ([ u ; x ])is a group isomorphism. In particular, Ψ is a functor.As both source and target of Ψ are groupoids, it suffices now to prove that I also forms a system of (representatives of) representatives of the isomorphismclasses on the right hand side.To see that I hits every isomorphism class at most once, assume ( u, x ) , ( v, y ) ∈ I represent isomorphic elements in ( E Inj(
A, ω ) × H X ) ϕ . Lemma 1.2.66 impliesthat there exists an h ∈ H with y.h = x and h − σ v ( k ) h = σ u ( k ) for all k ∈ K . .2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 55 Then ( w, x ) := ( v.h, y.h ) represents the same element as ( v, y ) in both C ϕK as wellas ( E Inj(
A, ω ) × H X ) ϕ , and one easily checks that σ w = σ u . Thus, [ u, x ] ∼ = [ w, x ] =[ v, y ] in C ϕK by Lemma 1.2.61, and hence ( u, x ) = ( v, y ) by definition of I .But I also covers the isomorphism classes of ( E Inj(
A, ω ) × H X ) ϕ : if ( v ; y )represents an arbitrary element of it and α ∈ M is again K -equivariant with ω r im α a complete K -set universe, then one easily checks that [ αv, v ; y ] is ϕ -fixed, so that it witnesses [ v ; y ] ∼ = [ αv ; y ]. In other words, we may assume that v misses a complete K -set universe. But in this case it defines an element of C K ,which is then by definition isomorphic in C K to some [ u, x ] with ( u, x ) ∈ I , andhence also in ( E Inj(
A, ω ) × H X ) ϕ by functoriality. (cid:3) Quotient vs. action categories.
Putting everything together we get:
Proof of Theorem 1.2.56.
As before let K ⊂ M be a universal subgroupand let ϕ : K → G be any group homomorphism. We have to show that (1 . . ϕ -fixed points.For this we consider the diagram of categories and functors ` ( u,x ) ∈ I B End C ϕK ([ u, x ]) ` ( u,x ) ∈ I B (cid:0) C H (im σ u ) ∩ Stab H ( x ) (cid:1) C ϕK (cid:0) (Inj( A, ω ) × H X )// M ) ϕ ( E Inj(
A, ω ) × H X ) ϕ ` Bτ Φ Ψ where the unlabelled arrow is the map in question (or rather: it induces it onnerves). The top path through this diagram sends f : [ u, x ] → [ u, x ] to [ u.τ ( f ) , u ; x ]while the lower one sends it to [ f u, u ; x ]. As u.τ ( f ) = f u by definition of τ , thesetwo agree, i.e. the above diagram commutes.We now observe that the top map is a weak homotopy equivalence by Proposi-tion 1.2.63, that the vertical maps are equivalences by Corollary 1.2.62 and Propo-sition 1.2.67, respectively, and that the lower left inclusion is a weak homotopyequivalence by Lemma 1.2.59. The claim now follows by 2-out-of-3. (cid:3) We will now study various change-of-group functorsfor the above models of G -global homotopy theory. This is particularly easy forthe models based on E M -actions, since here Lemma 1.1.40 and Lemma 1.1.41,respectively, specialize to: Corollary . Let α : H → G be any group homomorphism. Then α ! : E M - H -SSet H -global ⇄ E M - G -SSet G -global : α ∗ is a simplicial Quillen adjunction with homotopical right adjoint. (cid:3) Corollary . Let α : H → G be any group homomorphism. Then α ∗ : E M - G -SSet G -global injective ⇄ E M - H -SSet H -global injective : α ∗ is a simplicial Quillen adjunction. (cid:3) On the other hand, Propositions 1.1.42 and 1.1.43 imply:
Corollary . Let α : H → G be an injective group homomorphism. Then α ! : E M - H -SSet H -global injective ⇄ E M - G -SSet H -global injective : α ∗ is a Quillen adjunction. In particular, α ! is homotopical. (cid:3) G -GLOBAL HOMOTOPY THEORY Corollary . Let α : H → G be an injective group homomorphism. Then α ∗ : E M - G -SSet G -global ⇄ E M - H -SSet H -global : α ∗ is a simplicial Quillen adjunction. If ( G : im α ) < ∞ , then α ∗ is homotopical. (cid:3) Finally, Proposition 1.1.45 specializes to:
Corollary . Let α : H → G be any group homomorphism. Then α ! : E M - H -SSet → E M - G -SSet preserves weak equivalences between objects with free ker( α ) -action. (cid:3) The case of M -actions needs slightly more work: Corollary . Let α : H → G be any group homomorphism. Then α ! : M - H -SSet H -global ⇄ M - G -SSet G -global : α ∗ is a simplicial Quillen adjunction. Proof.
This follows as before for the H -universal and G -universal model struc-ture, respectively. Thus, it suffices by Proposition A.2.8 that α ∗ sends G -semistableobjects to H -semistable ones, which is obvious from the definition. (cid:3) Corollary . Let α : H → G be any group homomorphism. Then α ∗ : G - M -SSet G -global injective ⇄ H - M -SSet H -global injective : α ∗ is a simplicial Quillen adjunction. In particular, α ∗ is homotopical. Proof.
It is clear that α ∗ preserves injective cofibrations, so it only remainsto show that it is homotopical.However, while α ∗ commutes with E M × M –, it is not clear a priori that it isalso suitably compatible with E M× L M – since α ∗ usually does not preserve cofibrantobjects. Instead, we consider the commutative diagram M - G -SSet M - G -SSet semistable M - H -SSet E M - H -SSet semistable . –// M α ∗ α ∗ –// M If we equip the categories in the top and botom row with the G -global and H -globalweak equivalences, respectively, the horizontal arrows create weak equivalences byTheorem 1.2.49. On the other hand, the right vertical arrow is homotopical, as G -global weak equivalences between G -semistable objects are G -universal weakequivalences. Thus, also the left hand vertical arrow is homotopical. (cid:3) Corollary . Let α : H → G be an injective group homomorphism. Then α ! : H - M -SSet H -global injective ⇄ G - M -SSet H -global injective : α ∗ is a Quillen adjunction. In particular, α ! is homotopical. Proof.
By Proposition 1.1.42, α ! preserves injective cofibrations and it sends H -universal weak equivalences to G -universal ones. On the other hand, any H -global weak equivalence factors as an H -global acyclic cofibration followed by a H -universal weak equivalence. Since α ! sends the former to G -global weak equivalencesby Corollary 1.2.73, 2-out-of-3 implies that α ! is also homotopical. (cid:3) .2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 57 G -global homotopy theory vs. G -equivariant homotopy theory. As promised, we will now explain how to exhibit classical (proper) G -equivarianthomotopy theory as a Bousfield localization of our models of G -global homotopytheory. In fact, we will construct for both models a chain of four adjoint functors,in particular yielding two Bousfield localizations each. Before we can do any ofthis however, we need to understand a particular case of the ‘change of family’adjunction from Proposition 1.1.46 better: Definition . We define E (for ‘equivariant’) as the family of those graphsubgroups Γ H,ϕ of M × G such that H is universal and ϕ is injective .It will be crucial soon to understand the essential image of the left adjoint(1.2.14) λ : E M - G -SSet ∞E -w.e. → E M - G -SSet ∞ G -global w.e. of the localization functor. Let us give some intuition for this: on the left handside, we only remember the fixed points for injective ϕ : H → G . If now X isany E M - G -simplicial set, then λ ( X ) ϕ and X ϕ are weakly equivalent by abstractnonsense. On the other hand, if ψ : H → G is not necessarily injective, then λ ( X ) ψ should be somehow computable from the fixed points for injective homomorphisms.One of the first guesses might be that λ ( X ) ψ is weakly equivalent to X ¯ ψ where¯ ψ : H/ ker( ψ ) → G is the induced homomorphism (and we have secretly identified H/ ker( ψ ) with a universal subgroup of M isomorphic to it).This indeed turns out to be true. However, we of course do not want λ ( X ) ψ and λ ( X ) ¯ ψ to be merely abstractly weakly equivalent, but instead we want someexplicit and suitably coherent way to identify them. The following definition turnsthis heuristic into a rigorous notion: Definition . An E M - G -simplicial set X is called kernel oblivious if thefollowing holds: for any universal H, H ′ ⊂ M , any surjective group homomorphism α : H → H ′ , any arbitrary group homomorphism ϕ : H ′ → G , and any u ∈ M suchthat hu = uα ( h ) for all h ∈ H , the map u. – : X ϕ → X ϕα is a weak homotopy equivalence of simplicial sets. Theorem . The following are equivalent for an E M - G -simplicial set X : (1) X is kernel oblivious. (2) X lies in the essential image of (1 . . . (3) X is G -globally equivalent to an E -cofibrant X ′ ∈ E M - G -SSet . (4) In any cofibrant replacement π : X ′ → X in the E -model structure on E M - G -SSet , π is actually a G -global weak equivalence. The proof will require some preparations.
Lemma . Let f : X → Y be an E -weak equivalence in E M - G -SSet suchthat X and Y are kernel oblivious. Then f is a G -global weak equivalence. Proof.
Let H ⊂ M be universal and let ϕ : H → G be any group homomor-phism. We have to show that f ϕ is a weak homotopy equivalence.For this we choose a universal subgroup H ′ ⊂ M together with an isomorphism H ′ ∼ = H/ ker ϕ , which gives rise to a surjective homomorphism α : H → H ′ withker( α ) = ker( ϕ ). By the universal property of quotients, there then exists a unique¯ ϕ : H ′ → G with ¯ ϕα = ϕ ; moreover, ¯ ϕ is injective. G -GLOBAL HOMOTOPY THEORY We now appeal to Corollary 1.2.35 to find a u ∈ M such that hu = uα ( h ) forall h ∈ H , yielding a commutative diagram X ¯ ϕ X ¯ ϕα = X ϕ Y ¯ ϕ Y ¯ ϕα = Y ϕ . f ¯ ϕ u. – f ϕ u. – The horizontal maps are weak equivalences as X and Y are kernel oblivious, andthe left hand vertical map is a weak equivalence because f is an E -weak equivalence.Thus, also the right hand map is a weak equivalence as desired. (cid:3) Proposition . Let K ⊂ M be universal and let ψ : K → G be an injective homomorphism. Then the projection E M × ψ G → G/ im ψ is a G -globalweak equivalence (where the right hand side carries the trivial E M -action). Proof.
Let H ⊂ M be universal and let ϕ be any group homomorphism. Wewill prove that the induced map ( E M × ψ G ) ϕ → ( G/ im ψ ) ϕ = ( G/ im ψ ) im ϕ is anequivalence of categories.Full faithfulness amounts to saying that for all [ v ; g ] , [ v ; g ′ ] ∈ ( E M × ψ G ) ϕ Hom([ v ; g ] , [ v ′ ; g ′ ]) ∼ = ( ∗ if [ g ] = [ g ′ ] in G/ im ψ ∅ otherwise . For this we observe that Lemma 1.2.66 identifies the left hand side with { k ∈ K : g ′ ψ ( k ) = g, kσ v ( h ) k − = σ v ′ ( h ) for all h ∈ H } =: X. As ψ is injective, there is at most one k satisfying the first condition, so that inparticular | X | ≤
1. Moreover, if there exists a k ∈ X , then indeed [ g ′ ] = [ g ′ ψ ( k )] =[ g ]. On the other hand, if [ g ] = [ g ′ ], then we find a (unique) k ∈ K such that g ′ ψ ( k ) = g , and we claim that k ∈ X . But indeed, we have by Lemma 1.2.34 ϕ ( h ) g = gψ ( σ v ( h )) and ϕ ( h ) g ′ = g ′ ψ ( σ v ′ ( h ))for all h ∈ H . It follows that g ′ ψ ( σ v ′ ( h ) k ) = g ′ ψ ( σ v ′ ( h )) ψ ( k ) = ϕ ( h ) g ′ ψ ( k ) = ϕ ( h ) g = gψ ( σ v ( h )) = g ′ ψ ( k ) ψ ( σ v ( h )) = g ′ ψ ( kσ v ( h )) , where we have used twice that g = g ′ ψ ( k ). As ψ is injective, it follows that σ v ′ ( h ) k = kσ v ( h ), and hence k ∈ X as claimed.It remains to prove (essential) surjectivity. For this we let g ∈ G represent any ϕ -fixed point of G/ im ψ . This means by definition that there exists for each h ∈ H a κ ( h ) ∈ K such that ϕ ( h ) g = gψ ( κ ( h )). As ψ is injective, κ ( h ) is unique; one theneasily checks by the usual calculation that κ : H → K is a group homomorphism.Thus, Corollary 1.2.35 implies that there exists a u ∈ M such that hu = uκ ( h ) forall h ∈ H , and that [ u ; g ] is ϕ -fixed. This is then the desired preimage of [ g ]. (cid:3) Proposition . Let K ⊂ M be universal, let ψ : K → G be injective, andlet L be any simplicial set. Then ( E M × ψ G ) × L is kernel oblivious. Proof.
The kernel oblivious E M - G -simplicial sets are obviously closed un-der tensoring with simplicial sets, so that it suffices to prove that E M × ψ G is .2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 59 kernel oblivious. For this we let ϕ, α, u as in Definition 1.2.77. Then we have acommutative diagram ( E M × ψ G ) ϕ ( E M × ψ G ) ϕα ( G/ im ψ ) im ϕ ( G/ im ψ ) im ϕαu. – where the vertical arrows are the projections. By the previous proposition, they areequivalences of categories, and hence so is the top map by 2-out-of-3. This finishesthe proof. (cid:3) Proof of Theorem 1.2.78.
The implications (4) ⇒ (2) and (2) ⇒ (3) followimmediately from Remark 1.1.47. For the remaining implications we will use: Claim. If X is cofibrant in E M - G -SSet E , then X is kernel oblivious. Proof.
Let α, ϕ, u as in Definition 1.2.77. It suffices to verify that the natu-ral transformation u. – : (–) ϕ ⇒ (–) ϕα satisfies the assumptions of Corollary 1.1.8.But indeed, Condition (1) is an instance of the previous proposition, while all theremaining conditions are part of Lemma 1.1.33. △ The implication (3) ⇒ (1) now follows immediately from the claim. On theother hand, if X is kernel oblivious, and π : X ′ → X is any I -cofibrant replacement,then X ′ is kernel oblivious by the claim, so that π is a G -global weak equivalenceby Lemma 1.2.79. This shows (1) ⇒ (4), finishing the proof. (cid:3) We now consider the functor triv E M : G -SSet → E M - G -SSet that equips a G -simplicial set with the trivial E M -action. The equality of functors(1.2.15) (–) ϕ ◦ triv E M = (–) im ϕ (for any H ⊂ M and any homomorphism ϕ : H → G ) shows that this is homotopicalwith respect to the proper weak equivalences on the source and the G -global or E -weak equivalences on the target. Theorem 1.2.78 now yields: Corollary . The diagram (1.2.16) G -SSet ∞ proper E M - G -SSet ∞E E M - G -SSet ∞ G -globaltriv ∞ E M triv ∞ E M λ commutes up to canonical equivalence. Proof.
This is is obviously true if we replace λ by its right adjoint; in partic-ular, there is a natural transformation filling the above, induced by the unit η of λ ⊣ localization. To see that this is in fact an equivalence, it suffices (as λ is fullyfaithful) that the right hand arrow lands in the essential image of λ .By the above theorem this is equivalent to demanding that triv E M X be kerneloblivious. But indeed, if ϕ, α, u are as in Definition 1.2.77, then (triv E M ) ϕ =(triv E M ) ϕα by (1 . .
15) and u. – is actually the identity by definition. (cid:3) We can now prove the comparison between G -global and proper G -equivarianthomotopy theory: G -GLOBAL HOMOTOPY THEORY Theorem . The functor triv E M : G -SSet proper → E M - G -SSet G -global descends to a fully faithful functor on associated quasi-categories. This inducedfunctor admits both a left adjoint L ( E M\ –) as well as a right adjoint (–) R E M .Moreover, (–) R E M is a quasi-localization at the E -weak equivalences, and it in turnadmits another right adjoint R , which is again fully faithful. Proof.
We first observe that the adjunctions(1.2.17) E M\ – : E M - G -SSet G -global ⇄ G -SSet proper : triv E M and(1.2.18) E M\ – : E M - G -SSet E ⇄ G -SSet proper : triv E M are Quillen adjunctions with homotopical right adjoints by the equality (1 . . . .
17) induces the desired left adjoint of triv ∞ E M .To construct the right adjoint of triv E M it suffices to observe that whiletriv E M : G -SSet ⇄ E M - G -SSet : (–) E M is not a Quillen adjunction with respect to the above model structures, it becomesone once we use Corollary A.2.10 to enlarge the generating cofibrations of the G -global model structure as to contain all G/H × ∂ ∆ n ֒ → G/H × ∆ n for H ⊂ G finiteand n ≥ E M is left Quillen with respect to the injective G -global model structure on the target.To prove the remaining statements we will need: Claim.
The Quillen adjunction (1 . .
18) is a Quillen equivalence. In particular,triv ∞ E M : G -SSet ∞ proper → E M - G -SSet ∞E -w.e. is an equivalence of quasi-categories. Proof.
It suffices to prove the first statement. The equality (1 . .
15) showsthat the right adjoint preserves and reflects weak equivalences. It is thereforeenough that the ordinary unit η : X → triv E M ( E M\ X ), which is given by theprojection map, is an E -weak equivalence for any E -cofibrant E M - G -simplicial set X . By the above, both E M\ – as well as triv E M are left Quillen (after suitablyenlarging the cofibrations in the target), and they moreover commute with tensor-ing with simplicial sets. By Corollary 1.1.8 it therefore suffices that η is a weakequivalence for any E M × ψ G with K ⊂ M universal and ψ : K → G injective.An easy calculation shows that the projection E M × G → G descends to anisomorphism E M\ ( E M × ψ G ) → G/ im ψ , so we want to show that the projection E M× ψ G → G/ im ψ is an E -weak equivalence. But this is actually even a G -globalweak equivalence by Proposition 1.2.80, finishing the proof of the claim. △ We now contemplate the diagram (1 . .
16) from Corollary 1.2.82. By the aboveclaim together with Proposition 1.1.46 we then immediately conclude thattriv ∞ E M : G -SSet ∞ proper → E M - G -SSet ∞ G -global is fully faithful. Moreover, we deduce by uniqueness of adjoints that its right adjoint(–) R E M is canonically equivalent to the composition(1.2.19) E M - G -SSet ∞ G -global localization −−−−−−−→ E M - G -SSet ∞E -w.e.(triv ∞ E M ) − −−−−−−−→ G -SSet proper.2. G -GLOBAL HOMOTOPY THEORY VIA MONOID ACTIONS 61 (where the right hand arrow denotes any quasi-inverse) and hence indeed a quasi-localization at the E -weak equivalences. Finally, (1 . .
19) has a fully faithful rightadjoint by Proposition 1.1.46, given explicitly by G -SSet ∞ proper triv ∞ E M −−−−−→ E M - G -SSet ∞E -w.e. ρ −→ E M - G -SSet ∞ G -global which is then also right adjoint to (–) R E M , finishing the proof. (cid:3) Remark . In summary, we in particular have two Bousfield localizations L ( E M\ –) ⊣ triv E M and (–) R E M ⊣ R . In the ordinary global setting one is mostly interested in the analogue of the ad-junction triv E M ⊣ (–) R E M (cf. Remark 1.3.60) which is a ‘wrong way’ (i.e. right)Bousfield localization. Warning . Using the Quillen equivalence (1 . . . .
19) and hence of the right adjoint (–) R E M oftriv ∞ E M : namely, this can be computed by taking a cofibrant replacement withrespect to the E -model structure and then dividing out the left E M -action.In contrast to this, the left adjoint of triv ∞ E M is computed by taking a cofibrantreplacement with respect to the G -global model structure and then dividing out theaction. These two functors are not equivalent even for G = 1: namely, let H ⊂M be any non-trivial universal subgroup and consider the projection p : E M → E M /H . As E M and E M /H are both cofibrant in E M -SSet , we can calculatethe value of L ( E M\ –) at p simply by E M\ p : E M\ E M → E M\ E M /H , whichis a map between terminal objects, hence in particular an equivalence.On the other hand, p is not an E -weak equivalence (i.e. underlying weak equiv-alence), for example because π ( E M /H, [1]) ∼ = H (as H acts freely on the con-tractible space E M ). But (–) R E M is a quasi-localization at the E -weak equiva-lences and these are saturated as they are part of a model structure. Thus p R E M is not an equivalence, and in particular it cannot be conjugate to L ( E M\ –)( p ).For finite G , we can give easier descriptions of (–) R E M and R : Proposition . Assume G is finite and choose an injective homomor-phism i : G → M with universal image, inducing ( i, id) : G → E M × G . Then (1.2.20) ( i, id) ∗ : E M - G -SSet G -global ⇄ G -SSet : ( i, id) ∗ is a Quillen adjunction with homotopical left adjoint, and we have canonical equiv-alences (cid:0) ( i, id) ∗ (cid:1) ∞ ≃ (–) R E M and R ( i, id) ∗ ≃ R . Proof.
It is obvious from the definition that ( i, id) ∗ sends E -weak equivalences(and hence in particular G -global weak equivalences) to G -weak equivalences. More-over, it preserves cofibrations as the cofibrations on the right hand side are just theunderlying cofibrations. We conclude that (1 . .
20) is a Quillen adjunction and that( i, id) ∗ descends to E M - G -SSet ∞E → G -SSet ∞ .On the other hand, by Theorem 1.2.83 also λ descends accordingly and theresulting functor is quasi-inverse to the one induced by triv E M . The equality( i, id) ∗ ◦ triv E M = id G -SSet of homotopical functors then also exhibits the functorinduced by ( i, id) ∗ on E M - G -SSet ∞E as left quasi-inverse to the one induced bytriv E M which provides the first equivalence. The second is then immediate fromthe uniqueness of adjoints. (cid:3) G -GLOBAL HOMOTOPY THEORY Corollary . The homotopical functor triv M : G -SSet proper → M - G -SSet G -global descends to a fully faithful functor on associated quasi-categories. This inducedfunctor admits both a left adjoint L ( M\ –) as well as a right adjoint (–) R M . Thelatter is a quasi-localization at those f such that E M× L M f is an E -weak equivalence,and it in turn admits another right adjoint R which is again fully faithful.Moreover, the diagram (1.2.21) G -SSet ∞ proper E M - G -SSet ∞ G -global M - G -SSet ∞ G -globaltriv ∞ E M triv ∞M forget ∞ commutes up to canonical equivalence. It follows formally that the forgetful functor is also compatible with the re-maining functors in the two adjoint chains constructed above, and likewise for itsadjoints E M × L M – and R maps M ( E M , –). Proof.
We obviously have a Quillen adjunctiontriv M : G -SSet proper ⇄ G - M -SSet injective G -global : (–) M justifying the above description of the right adjoint (in fact, as in the proof ofTheorem 1.2.83 it would have been enough to enlarge the generating cofibrationsby the maps G/H × ∂ ∆ n ֒ → G/H × ∆ n for H finite and n ≥ M\ – : M - G -SSet G -global ⇄ G -SSet proper : triv M is a Quillen adjunction. For this we observe that this is true for the G -universalmodel structure on the left hand side (as triv M is then obviously right Quillen);in particular M\ – preserves cofibrations and triv M sends fibrant G -simplicial setsto G -universally fibrant M - G -simplicial sets. As this adjunction has an obvioussimplicial enrichment, it therefore suffices by Proposition A.2.8 and the character-ization of the fibrant objects provided in Corollary 1.2.40 that triv M has image inthe G -semistable M - G -simplicial sets, which is in fact obvious from the definition.To prove that (1 . .
21) commutes up to canonical equivalence, it suffices toobserve that the evident diagram of homotopical functors inducing it actually com-mutes on the nose. All the remaining statements then follow formally from thecommutativity of (1 . .
21) as before. (cid:3) G -global homotopy theory via diagram spaces Definition . We write I for the category of finite sets and injections andwe write I for the simplicial category obtained by applying E to the hom sets. An I -simplicial set is a functor I → SSet and we write I -SSet for the simplicially en-riched functor category Fun ( I, SSet ). An I -simplicial set is a simplicially enrichedfunctor I →
SSet and we write I -SSet := Fun ( I , SSet ).In the literature, the category I is also denoted by I or I (and unfortunatelyalso by I ); Sagave and Schlichtkrull proved that the category I -SSet models ordi-nary homotopy theory, see [ SS12 , Theorem 3.3]. .3. G -GLOBAL HOMOTOPY THEORY VIA DIAGRAM SPACES 63 We can also view I as a discrete analogue of the topological category L used inSchwede’s model of the unstable global homotopy category via L -spaces, see [ Sch18 ,Section 1.1]. In an earlier version, Schwede also sketched that I -spaces model globalhomotopy theory (with respect to finite groups), also cf. [ Hau19b , Section 6.1], forwhich we will give a full proof as Theorem 1.5.23.
We will now introduce G -global modelstructures on G - I - and G - I -simplicial sets and prove that they are equivalent tothe models from the previous section. For this we begin by constructing a suitablelevel model structure (as one does in constructing the global model structure on L -Top ) that we will then later Bousfield localize at the desired weak equivalences: Proposition . There is a (unique) model structure on G - I -SSet in whicha map f : X → Y is a weak equivalence or fibration if and only if for each finitegroup H and any finite H -set A with a free H -orbit, the map f ( A ) : X ( A ) → Y ( A ) is a weak equivalence or fibration, respectively, in the G H,G -model structure (with H acting via functoriality), i.e. for each group homomorphism ϕ : H → G the inducedmap X ( A ) ϕ → Y ( A ) ϕ is a weak equivalence or fibration, respectively.We call this the restricted level model structure . It is proper, combinatorial,and filtered colimits in it are homotopical. A possible set of generating cofibrationsis given by the maps ( I ( A, –) × ϕ G ) × ∂ ∆ n ֒ → ( I ( A, –) × ϕ G ) × ∆ n for n ≥ and H , A , and ϕ as above, and a set of generating acyclic cofibrations islikewise given by ( I ( A, –) × ϕ G ) × Λ nk ֒ → ( I ( A, –) × ϕ G ) × ∆ n with ≤ k ≤ n .The analogous model structure on G - I -SSet exists and has the same proper-ties; we again call it the restricted level model structure . A possible set of generatingcofibrations is given by the maps ( I ( A, –) × ϕ G ) × ∂ ∆ n ֒ → ( I ( A, –) × ϕ G ) × ∆ n for n ≥ and H , A , and ϕ as above, and a set of generating acyclic cofibrations islikewise given by ( I ( A, –) × ϕ G ) × Λ nk ֒ → ( I ( A, –) × ϕ G ) × ∆ n with ≤ k ≤ n .The simplicial adjunction (1.3.1) I × I – : G - I -SSet restricted ⇄ G - I -SSet restricted : forget . is a Quillen adjunction. We will obtain these model structures as an instance of a more general con-struction from [
Sch18 ] of ‘generalized projective model structure’ for suitable indexcategories. For this we will need the following terminology:
Definition . Let C be a complete and cocomplete closed symmetricmonoidal category. We say that a C -enriched category I has a dimension function if there exists a function dim : Ob( I ) → N such that(1) Hom( d, e ) is initital in C whenever dim( e ) < dim( d ).(2) If dim( d ) = dim( e ), then d ∼ = e . G -GLOBAL HOMOTOPY THEORY Example . Both I and I have dimension functions; a canonical choiceis the function sending a finite set A to its cardinality | A | . Moreover, if G is anydiscrete group, then composing with the projection I × BG → I or I × BG → I yields a dimension function on I × BG or I × BG , respectively. Proposition . Let I be a C -enriched category with dimension function dim , and assume we are given for each A ∈ I a model structure on the C -enrichedfunctor category Fun (End( A ) , C ) such that the following ‘consistency condition’holds: if dim( A ) ≤ dim( B ) , then any pushout of Hom(
A, B ) ⊗ End( A ) i is a weakequivalence in Fun (End( B ) , C ) for any acyclic cofibration i in Fun (End( A ) , C ) .Then there exists a unique model structure on Fun ( I , C ) such that a map f : X → Y is a weak equivalence or fibration if and only if f ( A ) : X ( A ) → Y ( A ) isa weak equivalence or fibration, respectively, for each A ∈ I .Moreover, if each Fun (End( A ) , C ) is cofibrantly generated with set of generat-ing cofibrations I A and set of generating acyclic cofibrations J A , then the resultingmodel structure is cofibrantly generated with set of generating cofibrations { Hom( A, –) ⊗ End( A ) i : A ∈ I , i ∈ I A } , and generating acyclic cofibrations { Hom( A, –) ⊗ End( A ) j : A ∈ I , j ∈ J A } . Proof.
This is a special case of [
Sch18 , Proposition C.23]. (cid:3)
Here, Hom( A, –) ⊗ End( A ) – is the left adjoint of the functor Fun ( I , SSet ) → Fun (End( A ) , SSet ) given by evaluation at A . For C = SSet , we can modelHom(
A, B ) ⊗ End( A ) X explicitly by the balanced product Hom(
A, B ) × End( A ) X ,i.e. the quotient of the ordinary product by the equivalence relation generated ineach simplicial degree by ( f σ, x ) ∼ ( f, X ( σ )( x )). Proof of Proposition 1.3.2.
We will only construct the model structure on G - I -SSet , the construction for G - I -SSet being similar. For this we want to appealto the above proposition (for C = SSet , I = BG × I ), so we have to check theconsistency condition. To this end we claim that in the above situation, the functor I ( A, B ) × Σ A – sends acyclic cofibrations of the G Σ A ,G -model structure to acycliccofibrations in the injective G Σ B ,G -model structure. But indeed, by cocontinuity itsuffices to check this on generating acyclic cofibrations, where this is obvious.This already shows that the model structure on G - I -SSet exists and that it iscofibrantly generated with generating (acyclic) cofibrations as claimed above. As G - I -SSet is locally presentable, we conclude that it is in fact combinatorial.The category G - I -SSet is enriched, tensored, and cotensored over SSet in theobvious way, and as the G Σ A ,G -model categories are simplicial, so is G - I -SSet .Similarly one proves right properness and the preservation of weak equivalencesunder filtered colimits.Moreover, forget admits a simplicial left adjoint (via simplicially left Kan ex-tension along I → I ), which we denote by I × I –. It is then obvious from thedefinition that forget is right Quillen, so that (1 . .
1) is a Quillen adjunction.It only remains to establish left properness, for which we observe that anystandard generating cofibration is a levelwise cofibration, and hence so is any cofi-bration of the restricted level model structure. The claim therefore follows fromCorollary 1.1.38. (cid:3) .3. G -GLOBAL HOMOTOPY THEORY VIA DIAGRAM SPACES 65 Construction . We write I for the category of all sets and injectionsand we write I for the simplicial category obtained by applying E to each hom set.Then the categories I and I are full subcategories of I and I , respectively. We willnow explain how to extend any I - or I -simplicial set to I or I , respectively:In the case of X : I → SSet we define X ( A ) = colim B ⊂ A finite X ( B )for any set A . If i : A → A ′ is any injection, we define the structure map X ( A ) → X ( A ′ ) as the map induced via the universal property of the above colimit by thefamily X ( B ) X ( i | B ) −−−−→ X ( i ( B )) → colim B ′ ⊂ A ′ finite X ( B ′ ) = X ( A ′ )for all finite B ⊂ A , where the unlabelled arrow is the structure map of the colimitfor the term indexed by i ( B ) ⊂ A ′ . We omit the trivial verification that thisfunctorial.In the case of a simplicially enriched X : I →
SSet we define the extensionanalogously on objects and morphisms. If now ( i , . . . , i n ) is a general n -cell of I ( A, A ′ ), then we define X ( i , . . . , i n ) as the composition∆ n × colim B ⊂ A finite X ( B ) ∼ = colim B ⊂ A finite ∆ n × X ( B ) → colim B ′ ⊂ A ′ finite X ( B ′ )where the isomorphism is the canonical one and the unlabelled arrow is inducedby X ( i | B , . . . , i n | B ) : ∆ n × X ( B ) → X ( i ( B ) ∪ i ( B ) ∪ · · · ∪ i n ( B )) for all finite B ⊂ A . We omit the easy verification that this defines a simplicially enrichedfunctor I →
SSet .One moreover easily checks that these become simplicially enriched functors bysending F : ∆ n × X → Y to the transformation given on a set A by∆ n × colim B ⊂ A finite X ( B ) ∼ = colim B ⊂ A finite ∆ n × X ( B ) colim F ( B ) −−−−−−−→ colim B ⊂ A finite Y ( B ) , and that with respect to this the canonical maps X ( A ) → colim B ⊂ A finite X ( B )for finite A define simplicially enriched natural isomorphisms; accordingly, we willfrom now on no longer distinguish notationally between the extension and theoriginal object. Remark . One can check that the above is a model for the (simpliciallyenriched) left Kan extension; however, we will at several points make use of theabove explicit description.
Remark . Simply by functoriality, the above construction lifts to providesimplicially enriched extension functors G - I -SSet → G - I -SSet and G - I -SSet → G - I -SSet for any group G . Lemma . If X is a G - I - or G - I -simplicial set, then its extension preservesfiltered colimits, i.e. if J is a filtered category and A • : J → I any functor, then thecanonical map colim j X ( A j ) → X (cid:0) colim j A j (cid:1) G -GLOBAL HOMOTOPY THEORY is an isomorphism. (Here it does not matter whether we form the colimit on theright hand side in Set or in I ). Proof.
The mapcolim j X ( A j ) = colim j colim B ⊂ A j finite X ( B ) → colim B ⊂ colim j A j finite X ( B ) = X (colim j A j )in question is given on the ( j, B )-term by X ( i ) : X ( B ) → X ( B ′ ) where i : A j → colim j A j =: A is the structure map and B ′ is the image of B under it.We define a map in the other direction as follows: a finite subset B ⊂ A is contained in the image of some i : A j → A , and we send X ( B ) via X ( i − )to X ( i − ( B )) in the ( j, i − ( B ))-term on the left hand side. We omit the easyverification that this is well-defined and inverse to the above map. (cid:3) Construction . Applying Construction 1.3.6 and then restricting alongthe inclusion B M → I or B ( E M ) → I , respectively, sending the unique object to ω , we get functorsev ω : G - I -SSet → M - G -SSet and ev ω : G - I -SSet → E M - G -SSet . Lemma . Let U be a complete H -set universe and let i : U → A be an H -equivariant injection. Then X ( i ) : X ( U ) → X ( A ) is a G H,G -weak equivalencefor any X ∈ G - I -SSet . Proof.
Let us first assume that A is countable. As U is a complete H -setuniverse by assumption, we can therefore find an H -equivariant injection j : A → U .Then X ( j ) is an H -homotopy inverse to X ( i ), as exhibited by the H -equivarianthomotopies X (id A , ij ) and X (id U , ji ), finishing the proof of the special case.In the general case we now observe that the map in question factors as X ( U ) ∼ = colim i ( U ) ⊂ B ⊂ A countable H -set X ( U ) ∼ −→ colim i ( U ) ⊂ B ⊂ A countable H -set X ( B ) ∼ = X ( A ) , where the left hand isomorphism uses that filtered categories have connected nerve,the second one uses the above special case levelwise, and the final one comes fromLemma 1.3.9. The claim follows immediately. (cid:3) Corollary . Let A be any set with an M -action and let i : ω → A be an M -equivariant injection. Then X ( i ) : X ( ω ) → X ( A ) is a G -global weak equivalenceof E M - G -simplicial sets for any G - I -simplicial set X (i.e., equivalently, a G -universal weak equivalence of M - G -simplicial sets) Proof.
This is obviously E M - G -equivariant, so that it suffices to apply theprevious lemma with U = ω and H varying over all universal subgroups of M . (cid:3) Definition . A map f : X → Y of G - I -simplicial sets (or G - I -simplicialsets) is called a weak equivalence at infinity if ev ω f = f ( ω ) : X ( ω ) → Y ( ω ) is a G -universal weak weak equivalence. Remark . The weak equivalences at infinity between G - I -simplicial setswill turn out to be the weak equivalences in a model structure on G - I -SSet mod-elling G -global homotopy theory, and we therefore also call them G -global weakequivalences . While also G - I -SSet models G -global homotopy theory, the weakequivalences are more subtle in this case (this is akin to the non-equivariant sit-uation as well as the closely related example of symmetric spectra). Here we are .3. G -GLOBAL HOMOTOPY THEORY VIA DIAGRAM SPACES 67 interested in the weak equivalences at infinity merely because they allow us to provemany results for G - I -SSet and G - I -SSet in a uniform manner. Lemma . Let f : X → Y be any restricted weak equivalence in G - I -SSet or G - I -SSet . Then f is also a weak equivalence at infinity. Proof.
It suffices to prove the second statement. Let H ⊂ M be universal;we have to show that X ( ω ) → Y ( ω ) is a G H,G -weak equivalence.Pick a free H -orbit F inside ω (which exists by universality). We then observethat we have a commutative diagramcolim F ⊂ A ⊂ ω finite H -set X ( A ) colim A ⊂ ω finite X ( A )colim F ⊂ A ⊂ ω finite H -set Y ( A ) colim A ⊂ ω finite Y ( A ) colim f ( A ) colim f ( A ) where the horizontal maps are induced from the inclusion of filtered posets { F ⊂ A ⊂ ω finite H -set } → { A ⊂ ω finite } . This is is cofinal: any finite subset A ⊂ ω is contained in the finite H -set F ∪ HA which is an element of the left hand side.Thus, the horizontal maps are isomorphisms, and it therefore suffices that theleft hand vertical map is a G H,G weak equivalence. But on this side, H simply actson each term of the colimit by functoriality, and each f ( A ) is a G H,G -weak equiva-lence with respect to this action. The claim follows as the G H,G -weak equivalencesare closed under filtered colimits. (cid:3)
Definition . A G - I -simplicial set (or G - I -simplicial set) X is called restricted static if for each finite group H , any group homomorphism ϕ : H → G ,and any H -equivariant injection i : A → B of H -sets such that A contains a free H -orbit, the induced map X ( A ) ϕ → X ( B ) ϕ is a weak equivalence. Lemma . Let
X, Y ∈ G - I -SSet or G - I -SSet be restricted static. Thena map f : X → Y is a weak equivalence at infinity if and only if it is a restrictedweak equivalence. Proof.
The implication ‘ ⇐ ’ holds without any assumptions, see Lemma 1.3.15.For the remaining implication, we let H be any finite group and A a finite H -setcontaining a free H -orbit; we have to show that f ( A ) is a G H,G -weak equivalence,for which we may assume without loss of generality that H literally is a universalsubgroup of M and that A is an H -subset of ω .We then consider the commutative diagram X ( A ) colim A ⊂ B ⊂ ω finite H -set X ( B ) X ( ω ) Y ( A ) colim A ⊂ B ⊂ ω finite H -set Y ( B ) Y ( ω ) f ( A ) colim f ( B ) ∼ = f ( ω ) ∼ = where the right hand portion comes from cofinality again, and the left hand hori-zontal maps are the structure maps of the colimit. By assumption, the right handvertical map is in particular a G H,G -weak equivalence, so it suffices that the left G -GLOBAL HOMOTOPY THEORY hand horizontal maps are also G H,G -weak equivalences. But this follows immedi-ately, as by assumption on X and Y all the transition maps of the above colimitsare G H,G -weak equivalences. (cid:3)
Lemma . Let
P XY Z fp qg be a pullback square in G - I -SSet such that q is a restricted fibration and g is a G -global equivalence. Then also f is a G -global equivalence. Proof.
As finite limits commute with filtered colimits and as limits commutewith each other, we get for any universal subgroup H ⊂ M and any group homo-morphism ϕ : H → G a pullback square P ( ω ) ϕ X ( ω ) ϕ Y ( ω ) ϕ Z ( ω ) ϕf ( ω ) ϕ p ( ω ) ϕ q ( ω ) ϕ g ( ω ) ϕ in SSet . The map g ( ω ) ϕ is a weak equivalence by definition, and we have to showthat also f ( ω ) ϕ is. For this it suffices by right properness of SSet that q ( ω ) ϕ is aKan fibration. But as before, after picking a free H -orbit F ⊂ ω , it can be identifiedwith the filtered colimit colim F ⊂ A ⊂ ω finite H -set q ( A ) ϕ of Kan fibrations, and hence is itself a Kan fibration as desired. (cid:3) The following is in particular an I -analogue of [ Sch18 , Theorem 1.1.10]:
Lemma . Let q : I → I be any enriched functor together with an enrichedtransformation e : incl ⇒ q . Define Q as the composition G - I -SSet extension −−−−−−→ G - I -SSet q ∗ −→ G - I -SSet , and η : id ⇒ Q as id ∼ = (restriction ◦ extension) e −→ ( q ∗ ◦ extension) = Q. Then η : X → QX is a G -global weak equivalence for any X ∈ G - I -SSet . Proof.
For a finite set A , the map η X ( A ) : X ( A ) → ( QX )( A ) is given by X ( A ) X ( e ) −−−→ X ( e ( A ) ⊂ q ( A )) → colim B ⊂ q ( A ) finite X ( B ) = ( QX )( A ) , where the right hand map is the structure map. When we pass to colimits, X ( ω ) = colim A ⊂ ω finite X ( A ) → colim A ⊂ ω finite X ( e ( A )) → colim A ⊂ ω finite colim B ⊂ q ( A ) finite X ( B ) , the monoid M acts on both sides via its canonical action on ω , and we have toshow that this is a G -universal weak equivalence. .3. G -GLOBAL HOMOTOPY THEORY VIA DIAGRAM SPACES 69 By the Fubini Theorem for colimits and a simple cofinality argument the doublecolimit on the right hand side computescolim A ⊂ ω finite ,B ⊂ q ( A ) finite X ( B ) ∼ = colim B ⊂ q ( ω ) finite X ( B ) = X ( q ( ω ))where we write q ( ω ) = colim A ⊂ ω finite q ( A ). The above identifications are M -equivariant if we again let M act via its action on ω everywhere; in particular theaction on the right hand side is given by functoriality.Under this identification, the map X ( ω ) → X ( q ( ω )) is induced by the M -equivariant injection colim A ⊂ ω finite e : ω → q ( ω ). Corollary 1.3.12 then impliesthat this is a G -universal weak equivalence, as desired. (cid:3) Construction . We apply the construction from the previous lemmato the following data:(1) The enriched functor q : I → I is defined as follows: a finite set A is sentto the set ω A of A -indexed tuples in ω . An injection i : A → B is sent tothe ‘extension by zero’ map i ! : ω A → ω B , i.e. if ( x a ) a ∈ A is an element of ω A , then i ! ( x • ) = ( y b ) b ∈ B with y b = ( x a if i ( a ) = b b / ∈ im i (observe that this is well-defined by injectivity of i ). Finally, the actionof q on higher cells is given in the unique possible way.(2) The enriched transformation e is given on a finite set A as the inclusion ofthe characteristic functions, i.e. for a ∈ A the element e ( a ) ∈ ω A is givenby e ( a ) a ′ = ( a = a ′ Q to the evaluation functor. For this we first constructsimplicially enriched functors M - G -SSet → G - I -SSet and E M - G -SSet → G - I -SSet that we both denote by (–)[ ω • ]. For G = 1 an analogous construction appearsas [ Sch19b , Construction 8.2], where it is denoted by ρ , also cf. [ Sch19b , Con-struction 3.3].
Construction . We write I ω ⊂ I for the full subcategory of countablyinfinite sets. Then the inclusion B M → I factors through an equivalence i : B M → I ω . We pick once and for all a retraction r ; this is then automatically quasi-inverseto i and we fix τ : ir ∼ = id with τ ω = id ω . The functor r uniquely extends to asimplicially enriched functor I ω → BE M (which we denote by the same symbol);this is automatically a retraction of the inclusion i , and τ is a simplicially enrichedisomorphism ir ∼ = id.If now X is any M - G -simplicial set, then we use the notation X [–] := X ◦ r . Wenow define X [ ω • ] to be the following G - I -simplicial set: if A = ∅ , then X [ ω A ] = X ( r ( ω A )) as above. The G -action is the obvious one and for any injection i : A → B we take the structure map to be X [ i ! ] : X [ ω A ] → X [ ω B ], i.e. it is given by applying G -GLOBAL HOMOTOPY THEORY X ◦ r to the ‘extension by zero map’ considered above. In the case of an E M -simplicial set, we moreover let an n -cell ( i , . . . , i n ) act by X [ i , . . . , i n ! ]. We remarkthat this means that as an underlying simplicial set X [ ω A ] = X and all of the abovestructure is given by acting with certain (inexplicit and mysterious) elements of M or E M .If X is merely a G - I -simplicial set, we now set X [ ω ∅ ] = X M and take thestructure maps X [ ω ∅ ] = X M → X = X [ ω A ] to be the inclusions. In the case of I we instead define X [ ω ∅ ] = X E M , i.e. its n -simplices are those σ ∈ X n such thatthe composition E M × ∆ n E M× σ −−−−−→ E M × X act −−→ X factors through the projection E M × ∆ n → ∆ n . The structure maps are againgiven by the inclusions and we choose all higher cells to be trivial. As all theremaining structure is given by acting with elements of M or E M , respectively,this is easily seen to be functorial.These assignments obviously become simplicially enriched functors M - G -SSet → G - I -SSet and E M - G -SSet → G - I -SSet when we send an n -cell f : ∆ n × X → Y to the transformation given in non-emptydegree by f itself and in degree ∅ by f M or f E M , respectively. Remark . There are two alternative perspectives on the above construc-tion, that we briefly sketch. For the arguments in this paper we will however onlybe interested in the above version of the construction.Firstly, in the M -case it can be identified with the right Kan extension along B M → I , postcomposed with restriction along I → I, A ω A , and similarly forthe E M -case (where a simplicially enriched right Kan extension appears).Secondly, there is a ‘coordinate free’ version of the above construction. Let usagain consider the M -case first: if A is any non-empty set, then Inj( ω A , ω ) is left M -isomorphic to M (by picking a bijection ω A ∼ = ω ); for A = ∅ , Inj( ω ∅ , ω ) ∼ = ω corepresents the simplicial subset of those simplices that are fixed by all u ∈ M with u (0) = 0. With a bit of work one can then produce a natural map from X [ ω • ] to the I -simplicial set that sends A ∈ I to maps M (Inj( ω A , ω ) , X ), where thefunctoriality in A is the obvious one. This map is an isomorphism in all positivedegrees and hence in particular a restricted weak equivalence. A similar argumentapplies in the E M -case. These ‘coordinate free’ descriptions are then analogousto [ Sch20 , construction after Proposition 3.5].Here is a version of Lemma 1.3.11 for the above construction, also cf. [
Sch19b ,Proposition 3.5-(ii)]:
Lemma . Let X be a G -semistable M - G -simplicial set (e.g. one thatcomes from an E M - G -simplicial set), let U be a complete H -set universe, let V beany other countable H -set and let i : U → V be any H -equivariant injection. Thenthe induced map X [ i ] : X [ U ] → X [ V ] is a G H,G -weak equivalence.
Proof.
We may assume without loss of generality that H is literally a universalsubgroup of M . The assumptions guarantee that both U and V are H -universes,so we max fix H -equivariant isomorphisms ψ U : U → ω and ψ V : V → ω (where ω is in both cases equipped with the tautological H -action). Then X [ i ] factors as X [ U ] X [ ψ U ] −−−−→ X [ ω ] X [ ψ V iψ − U ] −−−−−−−→ X [ ω ] X [ ψ − V ] −−−−−→ X [ V ] .3. G -GLOBAL HOMOTOPY THEORY VIA DIAGRAM SPACES 71 and the outer two maps in this are ( G × H )-equivariant isomorphisms. So it sufficesthat the middle map is an G H,G -weak equivalence. But by construction, it is justgiven by acting with u := ψ V iψ − U ∈ M on X = X [ ω ], and the H -equivarianceof all these maps translates into the statement that u centralizes H . Thus, this isindeed an H -weak equivalence by what it means to be G -semistable. (cid:3) Proposition . The functor (–)[ ω • ] : G - I -SSet → M - G -SSet admits asimplicial left adjoint l and these two together form a Quillen adjunction (1.3.2) l : G - I -SSet restricted ⇄ M - G -SSet G -universal : (–)[ ω • ] with fully homotopical right adjoint. Similarly, there exists a simplicial Quillenadjunction (1.3.3) ℓ : G - I -SSet restricted ⇄ E M - G -SSet : (–)[ ω • ] . For the proof we will need:
Lemma . Let A be an H -set containing a free H -orbit. Then ω A (the setof functions A → ω ) with left H -action via ( h.f )( a ) = f ( h − .a ) contains a complete H -set universe; if A is finite, then it is itself a complete H -set universe. Proof.
The finite case is [
Sch19b , Proposition 2.19]; the infinite case can bededuced from this by observing that if F ⊂ A is any H -subset, then the ‘extensionby zero’ map ω F → ω A is an H -equivariant injection; taking F to be a free H -orbityields the claim (the same trick also appears in loc. cit. ). (cid:3) Proof of Proposition 1.3.24.
We only prove the first statement, the argu-ment for the second being similar. The functor (–)[ ω • ] is obviously continuous andaccessible; as both source and target are locally presentable, the Special AdjointFunctor Theorem therefore implies the existence of a left adjoint. As the simplicialfunctor (–)[ ω • ] obviously preserves cotensors, this can be enhanced to a simpliciallyenriched adjunction.To finish the proof it suffices to show that (–)[ ω • ] preserves weak equivalencesas well as fibrations, i.e. whenever f : X → Y is a G -universal weak equivalence orfibration, then we have to show that f [ ω A ] ϕ is a weak equivalence or Kan fibra-tion, respectively, for each finite H -set A containg a free H -orbit and each grouphomomorphism ϕ : H → H ′ . But indeed, ω A is a complete H -set universe by theprevious lemma, and we may assume without loss of generality that H literally isa universal subgroup of M . We can therefore find an H -equivariant isomorphism ψ : ω A ∼ = ω showing that f [ ω A ] ϕ agrees up to conjugation by isomorphisms with f ϕ .The latter is by definition a weak equivalence or Kan fibration, respectively. (cid:3) Remark . The functor Q is not quite isomorphic to the composition(–)( ω )[ ω • ], but the only issue is in degree ∅ . Namely, if A is any nonempty finiteset, then the isomorphism τ : ω A → ω from the construction of (–)[ ω • ] induces( QX )( A ) = X ( ω A ) X ( τ ) −−−→ X ( ω ) = X ( ω )[ ω A ]and these are by definition compatible with all the relevant structure maps andmoreover natural in X .On the other hand, in degree ∅ we have ( QX )( ∅ ) = X ( ω ∅ ) (which is the valueat a point) whereas X ( ω )[ ω ∅ ] = X ( ω ) E M and these are in general incomparable. G -GLOBAL HOMOTOPY THEORY Remark . On G - I -simplicial sets, we can still define the functor Q andthe natural transformation η in the same way, but the conclusion of Lemma 1.3.19need not hold anymore. What is still true however (by the same argument), is thatthe maps X ( τ ) define isomorphisms QX ( A ) → X ( ω )[ ω A ] for all A = ∅ that arecompatible with all the relevant structure maps. Proposition . The functor (–)[ ω • ] defines homotopy equivalences ( G -semistable M - G -simplicial sets) → (restricted static G - I -simplicial sets) and ( E M - G -simplicial sets) → (restricted static G - I -simplicial sets) Moreover, ev ω restricts to homotopy inverses for these. Proof.
We only prove the first statement, the argument for the second onebeing similar. Let us first observe that X [ ω • ] is indeed restricted static for any G -semistable X : namely, if the finite H -set A contains a free H -orbit, then ω A is acomplete H -set universe by Lemma 1.3.25, so any H -equivariant inclusion i : A → B into another finite H -set induces a G H,G -weak equivalence X [ ω A ] → X [ ω B ] byLemma 1.3.23.We now define a natural transformation θ : id ⇒ (–)( ω )[ ω • ]; this actually worksfor any G - I -simplicial set X , not only the restricted static ones: in degree ∅ this isinduced by the structure map X ( ∅ ) → X ( ω ) of the defining colimit, which is easilyseen to factor through X ( ω )[ ω ∅ ] = X ( ω ) M . On the other hand, in degree A = ∅ this is given as the composition X ( A ) η −→ X ( ω A ) ∼ = X ( ω )[ ω A ] , where the isomorphism on the right hand side is the one from Remark 1.3.26. Weomit the easy verification that this is well-defined and natural.Now assume that X is actually restricted static, let H be finite, and let A bea finite H -set containing a free H -orbit. Then in particular A = ∅ and hence themap θ X ( A ) factors as the composition X ( A ) ∼ = X ( e ( A )) ι −→ colim e ( A ) ⊂ B ⊂ ω A finite H -set X ( B ) ∼ = X ( ω A ) ∼ = X ( ω )[ ω A ]where ι is the structure map, the left hand isomorphism is induced by the restrictionof e : A → ω A to its image, the penultimate isomorphism comes from the usualcofinality argument and the final one is induced by τ . It therefore suffices that ι isa G H,G -weak equivalence, which follows from the fact that all the structure maps ofthe (filtered) colimit are G H,G -weak equivalences by what it means to be restrictedstatic. This completes the proof that ev ω is right homotopy inverse.To see that it is also left homotopy inverse, we consider the following zig-zagfor each M - G -simplicial set Y : Y [ ω • ]( ω ) = colim A ⊂ ω finite Y [ ω A ] α −→ colim A ⊂ ω finite Y [ ω A ∐ ω ] β ←− colim A ⊂ ω finite Y [ ω ] = Y The transition maps on the two left hand colimits come from the extension by zeromaps ω A → ω B and the transition maps of the remaining colimit are trivial. Thegroup G acts by its action on Y everywhere. Moreover, M acts on all the colimitsvia its action on the index category (observe that this part of the action is trivialfor the rightmost colimit) and in addition on the middle colimit by its tautologicalaction on the ω -summand of Y [ ω A ∐ ω ] and finally by its given action on Y [ ω ] = Y .3. G -GLOBAL HOMOTOPY THEORY VIA DIAGRAM SPACES 73 for the final colimit. The maps α and β are given in each degree by the inclusions ω A ֒ → ω A ∐ ω ← ֓ ω . One immediately checks that they are well-defined and M - G -equivariant. Moreover, we can make the middle term into a functor in Y in theobvious way and with respect to this the maps α and β are clearly natural.So it only remains to prove that α and β are G -universal weak equivalenceswhenever Y is G -semistable. We prove this for α , the other argument being sim-ilar. For this let H ⊂ M be universal. We pick a free H -orbit F inside ω ; bythe same argument as before, α agrees up to conjugation by ( G × H )-equivariantisomorphisms with the map(1.3.4) colim F ⊂ A ⊂ ω finite H -set Y [ ω A ] → colim F ⊂ A ⊂ ω finite H -set Y [ ω A ∐ ω ]still induced in each degree by the inclusion ω A ֒ → ω A ∐ ω . However this is an H -equivariant injection from a complete H -set universe into a countable H -set.We therefore immediately conclude from Lemma 1.3.23 that (1 . .
4) is a G H,G -weakequivalence as desired. (cid:3)
As being restricted static is independent of the value at ∅ , we immediatelyconclude from the above together with Remark 1.3.26: Corollary . Let X ∈ G - I -SSet . Then QX is restricted static. (cid:3) We are now finally ready to construct model structures on G - I -SSet and G - I -SSet representing G -global homotopy theory. Theorem . There exists a (unique) model structure on G - I -SSet withthe same cofibrations as the restricted model structure and with the G -global weakequivalences as weak equivalences. We call this the restricted G -global model struc-ture . It is proper, combinatorial, and filtered colimits in it are combinatorial. More-over, its fibrant objects are precisely the restricted static restricted fibrant ones.Finally, the adjunction (1.3.5) ℓ : G - I -SSet restricted G -global ⇄ E M - G -SSet : (–)[ ω • ] is a Quillen equivalence with fully homotopical right adjoint, and there exists acanonical equivalence L ℓ ≃ (ev ω ) ∞ : G - I -SSet ∞ G -global → E M - G -SSet ∞ . Proof.
We will construct the model structure in two different ways; eachof these constructions will allow us to then easily deduce one half of the aboveadditional statements.We first want to apply Bousfield’s criterion (which we recall as Theorem A.2.1in the appendix) to the functor Q and the natural transformation η considered inConstruction 1.3.20. Let us prove that the assumptions are satisfied. Condition(Q1), i.e. that Q is homotopical, can easily be deduced from Proposition 1.3.28,but we will instead already prove something a bit stronger. Namely, we have forany map f : X → Y in G - I -SSet a sequence of logical equivalences(1.3.6) f is a G -global weak equivalence ⇔ Qf is a G -global weak equivalence ⇔ Qf is a restricted weak equivalencewhere the first equivalence uses that η is a levelwise G -global weak equivalenceby Lemma 1.3.19 together with the obvious 2-out-of-3-property for G -global weak G -GLOBAL HOMOTOPY THEORY equivalences, and the second one follows from Lemma 1.3.17 together with Corol-lary 1.3.29. As restricted weak equivalences are in particular G -global weak equiv-alences by Lemma 1.3.15, this in particular proves (Q1), but we also conclude thatthe Q -weak equivalences are indeed precisely the G -global weak equivalences.Let us prove (Q2) next. To see that Qη X is a restricted weak equivalence, itsuffices to apply (1 . .
6) to the G -global weak equivalence η X . On the other hand, η QX is a G -global weak equivalence, and its source and target are both restrictedstatic (Corollary 1.3.29 again), so that it is in fact a restricted weak equivalence byanother application of Lemma 1.3.17.Finally, (Q3) is a special case of Lemma 1.3.18. We conclude that the modelstructure exists and is proper. Let us now specialize Bousfield’s characterizationof the fibrant objects to our situation: namely, this tells us that a G - I -simplicialset X is fibrant in the new model structure if and only if it is restricted fibrantand η : X → QX is a restricted weak equivalence. We want to show that the latteris equivalent to X being restricted static. Indeed, η is a G -global equivalence asremarked above, and QX is restricted static by Corollary 1.3.29. As global equiv-alences between restricted static objects are already restricted weak equivalences,this proves the ‘ ⇐ ’ implication. For the other direction we observe that as in theprevious step, QX is restricted static and that restricted static objects are obvi-ously closed under restricted weak equivalences. This finishes the part of the proofrelying on Bousfield’s approach.We will now construct the model structure again, this time using Lurie’s cri-terion applied to the simplicial Quillen adunction (1 . . . .
5) is a Quillen equivalence. As this hasthe same cofibrations and the same fibrant objects as the previously constructedone, these two model structures agree by Proposition A.2.7.It only remains to construct the equivalence of quasi-functors L ℓ ≃ (ev ω ) ∞ .For this we first observe that ev ω indeed descends by definition of the G -globalweak equivalences. But then both are left inverse to the equivalence (–)[ ω • ] ∞ : for L ℓ this follows from (1 . .
5) being a Quillen equivalence, and in the other case thisfollows from Proposition 1.3.28. This completes the proof of the theorem. (cid:3)
Theorem . There exists a (unique) model structure on G - I -SSet whosecofibrations are the restricted cofibrations and whose weak equivalences are createdby I × L I – : Ho( G - I -SSet restricted ) → Ho( G - I -SSet restricted G -global ) . We call this the restricted G -global model structure and its weak equivalences the G -global weak equivalences. It is left proper, combinatorial, and filtered colimits init are homotopical. Moreover, its fibrant objects are precisely the restricted staticrestricted fibrant ones.Finally, there are simplicial Quillen equivalences (1.3.7) I × I – : G - I -SSet restricted G -global ⇄ G - I -SSet restricted G -global : forget .3. G -GLOBAL HOMOTOPY THEORY VIA DIAGRAM SPACES 75 and (1.3.8) l : G - I -SSet restricted G -global ⇄ M - G -SSet G -global : (–)[ ω • ] . Proof.
The diagram of homotopical functors E M - G -SSet (restr. static G - I -simplicial sets)( G -semistable M - G -simplicial sets) (restr. static G - I -simplicial sets) forget (–)[ ω • ] forget(–)[ ω • ] commutes up to the levelwise weak equivalence given on an E M - G -simplicial set X by the map forget( X [ ω • ]) → (forget X )[ ω • ] defined in non-empty degrees as theidentity and on ∅ as the inclusion X E M ֒ → X M . (It is a non-trivial fact, seeTheorem 1.4.15, that X E M and X M are actually equal, so that the above diagramin fact commutes on the nose, but we will not need this here.)Passing to associated quasi-categories, the two horizontal maps become equiva-lences (Proposition 1.3.28) and so does the left hand vertical map by Theorem 1.2.39together with Proposition A.1.15. Therefore, also the right hand vertical map in-duces an equivalence on associated quasi-categories.We now consider the sequence of simplicial Quillen adjunctions G - I -SSet restricted G - I -SSet restricted G - I -SSet restricted G -global , I× I –forget idid and we claim that the right derived functor of the right adjoint in the compositesimplicial Quillen adjunction is fully faithful with essential image the restrictedstatic G - I -simplicial sets. If we can prove this, Lurie’s criterion will provide uswith the desired model structure and prove that (1 . .
7) is a Quillen equivalence.But indeed, the right derived functor of the identity is fully faithful with essentialimage the restricted static G - I -simplicial sets. Using Proposition A.1.15 twicetogether with the previous observation then proves the claim.It only remains to prove that also the simplicial adjunction (1 . .
8) is a Quillenequivalence. For this we first observe that it is a Quillen adjunction as its lefthalf preserves cofibrations (as a consequence of Proposition 1.3.24) and its righthalf preserves fibrant objects (by the same lemma, together with the fact that itsends G -semistable M - G -simplicial sets to restricted static G - I -simplicial sets byProposition 1.3.28). The same argument as above then allows us to conclude fromProposition 1.3.28 that the right derived functor of (–)[ ω • ] is indeed an equivalence,finishing the proof. (cid:3) Proposition . Let f : X → Y be a morphism of G - I -simplicial sets suchthat f ( ω ) is a G -global weak equivalence. Then f is a G -global weak equivalence. Proof.
We have to show that for each restricted static T ∈ G - I -SSet theinduced map f ∗ : Hom Ho( G - I -SSet restricted ) ( B, T ) → Hom
Ho( G - I -SSet restricted ) ( A, T )is bijective. For this we observe that ev ω and (–)[ ω • ] are both homotopical withrespect to the restricted weak equivalences on G - I -SSet and the G -universal weak G -GLOBAL HOMOTOPY THEORY equivalences on M - G -SSet , so that they descend to the respective homotopy cate-gories. It follows automatically that the natural transformation θ : id ⇒ (–)[ ω • ] ◦ ev ω also descends; we conclude that the diagramHom( B, T ) Hom( B ( ω ) , T ( ω )) Hom( B, T ( ω )[ ω • ])Hom( A, T ) Hom( A ( ω ) , T ( ω )) Hom( A, T ( ω )[ ω • ]) ev ω f ∗ θ ∗ ◦ (–)[ ω • ] f ( ω ) ∗ f ∗ ev ω θ ∗ ◦ (–)[ ω • ] (where all the hom sets are taken in the respective homotopy categories) commutesand that the horizontal composites agree with θ ∗ . As T was assumed to be re-stricted static, we therefore see that the horizontal composites are isomorphisms.A straightforward diagram chase then shows that the left hand vertical map willbe an isomorphism provided that the middle one is. But T ( ω ) is G -semistable and f ( ω ) was assumed to be a G -global weak equivalence, finishing the proof. (cid:3) Remark . The above is analogous to the proof that π ∗ -isomorphisms ofsymmetric spectra are stable equivalences appearing as [ HSS00 , Theorem 3.1.11].We will give another independent proof of the proposition as part of Theo-rem 1.4.27, which will also (among other things) prove the converse statement,i.e. the G -global weak equivalences in G - I -SSet are precisely those f for which f ( ω ) is a G -global weak equivalence. However, we have decided to give the aboveproof now as it needs much less work than Theorem 1.4.27 and moreover allows usto draw several interesting consequences now that fit here naturally and which wewould have had to postpone otherwise. Corollary . The forgetful functor G - I -SSet → G - I -SSet is also ho-motopical with respect to the G -global weak equivalences on both sides. Proof.
Let f : X → Y be a G -global equivalence of G - I -simplicial sets,i.e. f ( ω ) is a G -global weak equivalence of E M - G -simplicial sets. Then the map(forget f )( ω ) of M - G -simplicial sets agrees up to conjugation by isomorphisms withforget( f ( ω )), hence it is also a G -global weak equivalence. The claim therefore fol-lows from Proposition 1.3.32. (cid:3) We now want to construct G -global modelstructures on G - I -SSet and G - I -SSet that have more cofibrations. Again we be-gin with some sort of level model structure which we will then localize: Proposition . There is a unique model structure on G - I -SSet where amap f : X → Y is a weak equivalence or fibration if and only if for each finite group H and each finite faithful H -set A the map f ( A ) : X ( A ) → Y ( A ) is a G H,G -weakequivalence respectively G H,G -fibration, respectively.We call this the strict level model structure and its weak equivalences the strictlevel weak equivalences . It is simplicial (with respect to the canonical enrichment),proper, combinatorial with generating cofibrations given by the maps ( I ( A, –) × ϕ G ) × ∂ ∆ n ֒ → ( I ( A, –) × ϕ G ) × ∆ n for n ≥ and H , A , and ϕ as above, and generating acyclic cofibrations given by ( I ( A, –) × ϕ G ) × Λ nk ֒ → ( I ( A, –) × ϕ G ) × ∆ n with ≤ k ≤ n . Moreover, filtered colimits in it are homotopical. .3. G -GLOBAL HOMOTOPY THEORY VIA DIAGRAM SPACES 77 The analogous model structure on G - I -SSet exists, is again simplicial, properand combinatorial with the analogous generating (acyclic) cofibrations, and filteredcolimits in it are homtopical. We also call it the strict level model structure andrefer to its weak equivalences as strict level weak equivalences .Finally, the simplicial Quillen adjunction I × I – : G - I -SSet strict ⇄ G - I -SSet strict : forget is a Quillen adjunction with homotopical right adjoint. Proof.
This is proven in the same way as Proposition 1.3.2 and we leave thedetails to the reader. (cid:3)
Definition . A G - I -simplicial set (or G - I -simplicial set) X is called static , if for all finite faithful H -sets A and all H -equivariant injections i : A → B into another finite H -set, the induced map X ( i ) : X ( A ) → X ( B ) is a G H,G -weakequivalence.One shows just as in Lemma 1.3.17:
Lemma . Let f : X → Y be a map of static G - I -simplicial sets or G - I -simplicial sets. Then f is a G -global equivalence if and only if f is a strict weakequivalence. (cid:3) Theorem . There is a unique model structure on G - I -SSet in which amap is a cofibration if and only if it is one in the strict model structure, and whoseweak equivalences are the G -global weak equivalences.We call this the G -global model structure . It is combinatorial, simplicial,proper, and filtered colimits in it are homotopical. Moreover, its fibrant objectsare precisely the static strictly fibrant G - I -simplicial sets.Finally, the identity constitutes a Quillen equivalence (1.3.9) id : G - I -SSet restricted G -global ⇄ G - I -SSet G -global : id . For the proof we will use:
Lemma . G -global weak equivalences in G - I -SSet or G - I -SSet are sta-ble under pushout along injective cofibrations. Proof.
Pushouts along injective cofibrations preserve restricted level weakequivalences by Corollary 1.1.38, and as in the proof of Theorem 1.2.41 one candeduce from this that also G -global weak equivalences are stable under pushoutsalong injective cofibrations. (cid:3) Proof of Theorem 1.3.38.
We construct the model structure in two differ-ent ways. Let us first consider the functor q : I → I that is given on objects by A ω ( A + ) ∞ , on morphisms via extension by zero, and then on higher cells in theunique possible way. We want to apply Bousfield’s criterion (Theorem A.2.1) to thefunctor Q and the natural transformation η obtained by applying the constructionfrom Lemma 1.3.19 to q and the enriched natural transformation e : id ⇒ q givenby sending a ∈ A to the characteristic function of ( a, a, . . . ) ∈ ( A + ) ∞ .We claim that QX is static. Indeed, it suffices by Lemma 1.3.25 together withLemma 1.3.11 that ( A + ) ∞ contains a free H -orbit as soon as A is a faithful H -set.But if a , . . . , a n is any enumeration of the elements of A , then H acts freely on( a , . . . , a n , ∗ , ∗ , . . . ) ∈ ( A + ) ∞ , where ∗ denotes the extra point added to A (note G -GLOBAL HOMOTOPY THEORY that this is indeed necessary as A could be empty). With this established, it followsjust as in the proof of Theorem 1.3.30 that the assumptions of Bousfield’s criterionare satisfied, that the Q -weak equivalences are precisely the G -global weak equiva-lences, and that moreover η is a strict weak equivalence if and only if X is static.We conclude that the desired model structure exists, is proper, and that its fibrantobjects are as described above. The adjunction (1 . .
9) is a Quillen equivalence asthe restricted G -global model structure has fewer generating cofibrations as the G -global one, while both sides have the same weak equivalences.We already know that the G -global weak equivalences are closed under allfiltered colimits. To check that the G -global model structure is combinatorial, weconstruct it in a different way: we apply Corollary A.2.11 to the restricted G -global and the strict level model structure (we are indeed allowed to do this, as the G -global model structure is left proper as seen above).To finish the proof it therefore suffices that this model structure is simplicial,for which we have to verify the pushout product axiom. That a pushout productof cofibrations is again a cofibration follows immediately from the correspondingstatement for the strict level model structure. For the remaining part it againsuffices (using Lemma 1.3.39) that the G -global weak equivalences are stable undertensoring with simplicial sets and under products with arbitrary G - I -simplicial sets,both of which are trivial to prove. (cid:3) For later use we record:
Theorem . There is a unique cofibrantly generated model structure on G - I -SSet with weak equivalences the G -global weak equivalences and generatingcofibrations the maps I ( A, –) × ϕ G × ∂ ∆ n ֒ → I ( A, –) × ϕ G × ∆ n for n ≥ , finite groups H , homomorphisms ϕ : H → G and non-empty finite faith-ful H -sets A . We call this the positive G -global model structure . It is combinato-rial, simplicial, proper, and filtered colimits in it are homotopical. Moreover, a G - I -simplicial set X is fibrant if and only if X ( A ) is fibrant in the G Σ A ,G -equivariantmodel structure for any non-empty A , and X ( i ) : X ( A ) → X ( B ) is a G Σ A ,G -weakequivalence for any injection i : A → B of non-empty finite sets.Finally, the identity adjunction G - I -SSet positive G -global ⇄ G - I -SSet G -global is a Quillen equivalence. (cid:3) The proof of the theorem is completely analogous to the previous argumentsand we leave it to the reader.
Lemma . Let f : X → Y be a cofibration in the G -global positive modelstructure. Then f ( ∅ ) : X ( ∅ ) → Y ( ∅ ) is an isomorphism. Proof.
The class of such maps is obviously closed under retracts and all col-imits. Thus, it suffices to prove the claim for each generating cofibration i . Butin this case both source and target are obviously empty in degree ∅ , in particular i ( ∅ ) is an isomorphism. (cid:3) Warning . One might be tempted now to try to prove the correspond-ing theorems for G - I -SSet . However, there is an issue here: if we apply Lurie’scriterion (Theorem A.2.5) to the adjunction I × I – ⊣ forget again, then it is not apriori clear what the obtained weak equivalences are, since we are deriving I × I – .3. G -GLOBAL HOMOTOPY THEORY VIA DIAGRAM SPACES 79 with respect to a different model structure now. If on the other hand, we insteadused Corollary A.2.11 again, then this would have the correct weak equivalences,but we would have no control over the fibrant objects.One can in fact show that applying Lurie’s criterion yields the correct weakequivalences, but doing that at this point would be non-trivial. As the proof willbecome much easier once we have understood the G -global weak equivalences better(Theorem 1.4.27 below), we have decided to postpone this.Finally, we come to injective model structures: Theorem . There exists a unique model structure on G - I -SSet whosecofibrations are the levelwise injections and whose weak equivalences are the G -global weak equivalences. We call this the injective G -global model structure . Itis combinatorial, proper, simplicial (with respect to the obvious enrichment), andfiltered colimits in it are homotopical. Proof.
As we have seen in the proof of Proposition 1.3.2, the cofibrationsof the restricted G -global model structure are in particular injective cofibrations.On the other hand, pushouts along injective cofibrations preserve G -global weakequivalences by Lemma 1.3.39. Corollary A.2.11 therefore shows that the modelstructure exists, that it is combinatorial and left proper, and that filtered colimitsin it are homotopical.It only remains to verify the pushout product axiom for the simplicial tensoring,which in turn follows immediately for the pushout product axiom for SSet and forthe injective model structure on E M - G -SSet . (cid:3) Theorem . There exists a unique model structure on G - I -SSet whosecofibrations are the levelwise injections and whose weak equivalences are the G -global weak equivalences. We call this the injective G -global model structure . It iscombinatorial, left proper, simplicial (with respect to the obvious enrichment), andfiltered colimits in it are homotopical. Moreover, the simplicial adjunction forget: G - I -SSet injective G -global ⇄ G - I -SSet injective G -global : maps I ( I , –) is a Quillen equivalence. Proof.
The first part is analogous to the proof of the previous theorem, exceptfor the verification of the pushout product axiom, where one instead argues as inthe proof of Theorem 1.2.41.For the final statement, we observe that the forgetful functor preserves weakequivalences by Corollary 1.3.34 and injective cofibrations for trivial reasons, so itis left Quillen. It is then a Quillen equivalence by Theorem 1.3.31. (cid:3)
Schwede gives a characterizationof the cofibrations in the model structures of Proposition 1.3.5, that we now wantto recall for I -simplicial sets. For this we need the following notion: Construction . For any X ∈ G - I -SSet and any finite set A we write L A ( X ) = colim B ( A X ( B )(where { B ( A } is viewed as a poset under inclusion) and call it the A th latchingobject . The structure maps X ( B ) → X ( A ) assemble into a morphism ℓ A : L A ( X ) → X ( A ) that we call the A th latching map . We remark that we can make L A into afunctor by functoriality of colimits and that with respect to this ℓ A is natural. G -GLOBAL HOMOTOPY THEORY Remark . The latching object is usually defined as the analogous colimitover the latching category ∂ ( I ↓ A ) defined as the full subcategory of the slice I ↓ A on all objects B → A that are not isomorphisms. The latching map is then againdefined by the maps X ( B ) → X ( A ).However, this is equivalent to the above as the evident inclusion { B ( A } → ∂ ( I ↓ A ) is an equivalence of categories (and hence in particular cofinal) by astraight-forward computation. Remark . One can define the latching objects also more generally forenriched indexing categories by using an enriched left Kan extension, see [
Sch18 ,Construction C.13 and Definition C.15]. In the case of a discrete indexing categorythis specializes to the above by Kan’s pointwise formula.The characterization from [
Sch18 , Proposition C.23] then specializes to:
Lemma . (1) A map f : X → Y in G - I -SSet is a cofibration inthe strict level model structure if and only if for each finite set A the map (1.3.10) X ( A ) ∐ L A ( X ) L A ( Y ) ( f ( A ) ,ℓ A ) −−−−−−→ Y ( A ) (where the pushout is taken over the maps ℓ A : L A ( X ) → X ( A ) and L A ( f ) ) is a cofibration in the G Σ A ,G -model structure on ( G × Σ A )-SSet .In particular, X is cofibrant in the strict model structure if and onlyif for each finite set A the map ℓ A : L A ( X ) → X ( A ) is a cofibration in theabove model category. (2) The map f is a cofibration in the restricted level model structure if andonly if (1 . . is a cofibration in the model structure on ( G × Σ A )-SSet with respect to the family of those graph subgroups Γ H,ϕ such that thetautological action of H ⊂ Σ A on A has a free H -orbit.In particular, X is cofibrant if and only if all the latching maps arecofibrations in the above model structure. (cid:3) With this description at hand we can now prove:
Proposition . The forgetful functor G - I -SSet strict → G - I -SSet strict (which is right Quillen by Proposition 1.3.35) is also left Quillen. Proof.
The forgetful functor admits a right adjoint given by simplicially en-riched right Kan extension along I → I (or alternatively by the Special AdjointFunctor Theorem); we denote this right adjoint by maps I ( I , –). As forget is homo-topical, it therefore only remains to prove that it sends generating cofibrations tocofibrations. Claim.
Let
A, B be finite sets and let n ≥
0. Then the latching mapcolim C ( B I ( A, C ) n +1 → I ( A, B ) n +1 is injective. A tuple ( f , . . . , f n ) ∈ I ( A, B ) n +1 is not contained in the image if andonly if it is jointly surjective. Proof.
Let ( f , . . . , f n ) be a family of injections A → C and let ( f ′ , . . . , f ′ n )be a family of injections A → C ′ for proper subsets C, C ′ ( B , such that both aresent to the same element of I ( A, B ) n +1 , i.e. for each a ∈ A and i = 0 , . . . , nC ∋ f i ( a ) = f ′ i ( a ) ∈ C ′ . .3. G -GLOBAL HOMOTOPY THEORY VIA DIAGRAM SPACES 81 We conclude that f i and f i both factor through the same injection f ′′ i : A → C ∩ C ′ . But then obviously ( f , . . . , f n ) represents the same element of the colimit as( f ′′ , . . . , f ′′ n ), and so does ( f ′ , . . . , f ′ n ), finishing the proof of injectivity.It remains to prove the characterization of the image. Indeed, if ( f , . . . , f n ) ∈ I ( A, C ) n +1 for C ( B then im( f ) ∪ · · · ∪ im( f n ) ⊂ C , so that they are not jointlysurjective viewed as maps to B . On the other hand, if ( g , . . . , g n ) ∈ I ( A, B ) n +1 isnot jointly surjective, then C := im( g ) ∪ · · · ∪ im( g n ) is a proper subset of B . Bydefinition, each g i factors through a g ′ i : A → C and then ( g ′ , . . . , g ′ n ) is the desiredpreimage. △ Now let A be a finite faithful H -set and let B be any finite set. We can thenview I ( A, B ) n +1 as a (Σ B × H )-set. We claim that the isotropy group of any ( f , . . . , f n ) ∈ I ( A, B ) n +1 is a graph subgroup (for some group homomorphismfrom some subgroup of Σ B to H ). This amounts to saying that no 1 = h ∈ H fixes( f , . . . , f n ). But indeed, if h = 1, then we can find by faithfulness of the action an a ∈ A with h − .a = a . But then ( h.f )( a ) = f ( h − .a ) = f ( a ) by injectivity of f ,as desired.We are now ready to finish the proof of the proposition: let A be any finitefaithful H -set, let ϕ : H → G be a group homomorphism, and let B be any finiteset. The above claim tells us that the latching map(1.3.11) i : colim C ( B I ( A, C ) → I ( A, B )is a levelwise injection, and the argument from the previous paragraph tells us inparticular that any simplex not in the image has isotropy a graph subgroup ofΣ B × H . This precisely means that (1 . .
11) is a cofibration for the G Σ B ,H -modelstructure on (Σ B × H )-SSet .The functor(1.3.12) – × ϕ G : (Σ B × H )-SSet → (Σ B × G )-SSet is left adjoint to restriction along Σ B × ϕ . The latter is obviously right Quillen withrespect to the graph model structures on either side, hence (1 . .
12) is left Quillenwith respect to these model structures. But the B th latching map ℓ B of I ( A, –)factors ascolim C ( B ( I ( A, C ) × ϕ G ) ∼ = (cid:0) colim C ( B I ( A, C ) (cid:1) × ϕ G i × ϕ G −−−−→ I ( A, B ) × ϕ G (where the unlabelled isomorphism uses that – × ϕ G commutes with colimits), hence ℓ B is indeed a cofibration for the graph model structure on (Σ B × G )-SSet . Bythe previous lemma we therefore conclude that I ( A, –) × ϕ G is a cofibrant G - I -simplicial set. As the strict model structure is simplicial, this implies that I ( A, –) × ϕ G × ∂ ∆ n ֒ → I ( A, –) × ϕ G × ∆ n is a strict cofibration of G - I -simplicial sets. Since these are precisely the images ofthe generating cofibrations of G - I -SSet , this finishes the proof. (cid:3) Proposition . The forgetful functor G - I -SSet restr. → G - I -SSet restr. (which is right Quillen by Proposition 1.3.2) is also left Quillen. Proof.
By the same arguments as in the previous proposition, it suffices thatfor any finite H -set A containg a free H -orbit and any finite set B the latching map i : colim C ( B I ( A, C ) → I ( A, B ) G -GLOBAL HOMOTOPY THEORY is a cofibration in the model structure on (Σ B × H )-SSet with respect to thefamily of those graph subgroups Γ K,ψ such that the tautological action on B by K ⊂ Σ B contains a free K -orbit. We already know by the previous proposition,that this map is injective and an n -simplex ( f , . . . , f n ) is not in the image preciselyif the f i are jointly surjective. Moreover, we have seen there, that the isotropy groupof ( f , . . . , f n ) is some graph subgroup Γ K,ψ , so it only remains to prove that K has the desired properties. Claim.
The homomorphism ψ : K → H is injective. Proof.
Indeed, assume ψ ( k ) = 1 for k = 1. As K acts faithfully on B , wefind a b ∈ B with k.b = b . On the other hand, we find by joint surjectivity an i ∈ { , . . . , n } and an a ∈ A with b = f i ( a ). But then( k, ψ ( k )) . ( f , . . . , f n ) = ( k, . ( f , . . . , f n ) = ( k. – ◦ f , . . . , k. – ◦ f n ) , so evaluating the i -th entry at a shows that this does not agree with ( f , . . . , f n ),contradicting the assumptions. Thus, ψ is indeed injective. △ With this established, we can now find a free K -orbit in B : by assumption,there is an a ∈ A such that the orbit Ha is free. Then k. ( f ( a )) = (cid:0) ( k, ψ ( k )) .f (cid:1) ( ψ ( k ) .a ) = f ( ψ ( k ) .a ) , and this is different from f ( a ) for k = 1 by injectivity of ψ and f . In other words, Kf ( a ) is a free orbit, completing the proof. (cid:3) Corollary . We have a simplicial Quillen equivalence forget : G - I -SSet restricted G -global ⇄ G - I -SSet restricted G -global : maps I ( I , –) . Proof.
Let us first show that this is a Quillen adjunction. Indeed, as the re-stricted global cofibrations agree with the restricted cofibrations, Proposition 1.3.50tells us that the forgetful functor preserves cofibrations. On the other hand, it ishomotopical by Corollary 1.3.34, so it is in particular left Quillen.Finally, as forget is homotopical, its right derived functor agrees with its leftderived one. Therefore, Theorem 1.3.31 implies that the above is indeed a Quillenequivalence, finishing the proof. (cid:3)
Once we have established the G -global model structure on G - I -SSet we willalso get the corresponding statement for those, see Corollary 1.4.30. We will now discuss change of group functors for ourmodels based on I -simplicial sets. Lemma . Let α : H → G be any group homomorphism. Then (1.3.13) α ! : H - I -SSet H -global ⇄ G - I -SSet G -global : α ∗ is a simplicial Quillen adjunction with fully homotopical right adjoint, and likewisefor the restricted or positive model structures on either side. Proof.
We will prove the first statement, the other ones can be proven anal-ogously. One immediately checks that α ! ⊣ α ∗ is a Quillen adjunction with respectto the strict level model structures. On the other hand, α ∗ obviously sends static G - I -simplicial sets to static H - I -simplicial sets, so that also (1 . .
13) is a Quillenadjunction by Proposition A.2.8. .3. G -GLOBAL HOMOTOPY THEORY VIA DIAGRAM SPACES 83 To see that α ∗ is homotopical we observe that in the commutative diagram G - I -SSet G -global G - E M -SSet G -global H - I -SSet H -global H - E M -SSet H -globalev ω α ∗ α ∗ ev ω the horizontal arrows preserve and reflect weak equivalences by definition whilethe right hand vertical arrow obviously preserves weak equivalences. The claimfollows. (cid:3) Corollary . Let α : H → G be any group homomorphism. Then α ∗ : G - I -SSet G -global injective ⇄ H - I -SSet H -global : α ∗ is a simplicial Quillen adjunction. Proof.
It is obvious that α ∗ preserves injective cofibrations and it is moreoverhomotopical by the previous lemma, hence left Quillen. (cid:3) Lemma . Let α : H → G be an injective group homomorphism. Then α ! : H - I -SSet H -global injective ⇄ G - I -SSet G -global injective : α ∗ is a simplicial Quillen adjunnction. In particular, α ! is homotopical. Proof.
We may assume without loss of generality that H is a subgroup of G and that α is its inclusion, in which case α ! can be modelled by applying G × H –levelwise. We immediately see that α ! preserves injective cofibrations. To finish theproof it suffices now to show that it is also homotopical for which we consider H - I -SSet H - E M -SSet G - I -SSet G - E M -SSet . α ! ev ω α ! ev ω We claim that this commutes up to isomorphism. Indeed, as α ! is cocontinuous,there is a canonical G -equivariant isomorphism filling this, and one easily checksthat this isomorphism is also E M -equivariant.But the horizontal arrows in the above diagram preserve and reflect weak equiv-alences by definition and the right hand arrow is homotopical by Corollary 1.2.70,so the claim follows immediately. (cid:3) Lemma . Let α : H → G be an injective group homomorphism. Then thesimplicial adjunction α ∗ : G - I -SSet G -global ⇄ H - I -SSet G -global : α ∗ is a Quillen adjunction. If ( G : im α ) < ∞ , then α ∗ is homotopical. Proof.
Let us first show that this is a Quillen adjunction. We already knowthat α ∗ is homotopical, so it suffices to show that the above is a Quillen adjunctionfor the strict level model structures, which follows in turn by applying Proposi-tion 1.1.43 levelwise.Finally, if ( G : im α ) < ∞ , then α ∗ is non-equivariantly just given by a finiteproduct. Using that filtered colimits commute with finite products, one concludessimilarly to the argument from the previous lemma that α ∗ commutes with ev ω . G -GLOBAL HOMOTOPY THEORY The claim follows as α ∗ : E M - H -SSet → E M - G -SSet is homotopical by Corol-lary 1.2.71. (cid:3) As an application of the calculus just developed we can now prove:
Proposition . Let X be fibrant in the injective G -global model structureon G - I -SSet and let i : A → B be a G -equivariant injection of (not necessarily fi-nite) G -sets. Then X ( i ) : X ( A ) → X ( B ) is a proper G -equivariant weak equivalence(with respect to the diagonal G -action). Proof.
Fix a finite subgroup H ⊂ G ; we have to show that X ( i ) H : X ( A ) H → X ( B ) H is a weak homotopy equivalence. By 2-out-of-3 we may assume without lossof generality that A = ∅ , and filtering B by its finite H -subsets we may assumethat B itself is finite.We now observe that X is also fibrant in the H -global injective model structureby Lemma 1.3.54. On the other hand, by the Yoneda Lemma X ( i ) H agrees up toconjugation by isomorphisms withmaps H ( p, X ) : maps H ( ∗ , X ) → maps H ( I ( B, –) , X )where p : I ( B, –) → ∗ is the unique map.As the injective H -global model structure is simplicial, it therefore suffices that p is an H -global weak equivalence, which by definition amounts to saying that E Inj(
B, ω ) → ∗ is an H -global weak equivalence of E M - H -simplicial sets. Indeed,if K ⊂ M is universal and ϕ : K → H any group homomorphism, then (cid:0) E Inj(
B, ω ) (cid:1) ϕ ∼ = E (cid:0) Inj(
B, ω ) ϕ (cid:1) and so it suffices to find a ϕ -fixed point of Inj( B, ω ). For this we observe that ϕ ∗ B is a finite K -set and hence admits a K -equivariant embedding i into the complete K -set universe ω . The K -equivariance of i precisely means that i ( ϕ ( k ) .x ) = ki ( x ),i.e. i is a ϕ -fixed point. This finishes the proof. (cid:3) G -equivariant homotopy theory. We can nowprove the analogues of the results of Section 1.2.6 for G - I -simplicial sets: Corollary . The homotopical functor const: G -SSet proper → G - I -SSet G -global induces a fully faithful functor on associated quasi-categories. This induced functoradmits both a left adjoint L colim I as well as a right adjoint R ev ∅ . The latteris a quasi-localization at those f such that f ( ω ) is an E -weak equivalence of G - E M -simplicial sets; it in turn admits another right adjoint R , which is again fullyfaithful.Finally, the diagram (1.3.14) G -SSet ∞ proper G - I -SSet ∞ G -global G - E M -SSet ∞ G -globalconst ∞ triv ∞ E M (ev ω ) ∞ commutes up to canonical equivalence. .3. G -GLOBAL HOMOTOPY THEORY VIA DIAGRAM SPACES 85 Proof.
The adjunctionconst : G -SSet proper ⇄ G - I -SSet G -global injective : ev ∅ is easily seen to be a Quillen adjunction, providing the above description of theright adjoint. Similarly, for the left adjoint we want to prove that(1.3.15) colim I : G - I -SSet G -global restricted ⇄ G -SSet proper : constis a Quillen adjunction. For this we first observe that this is a Quillen adjunctionwhen we equip the left hand side with the strict model structure (as const is thenobviously right Quillen). In particular, colim I preserves G -global (i.e. strict) cofi-brations and const sends fibrant G -simplicial sets to strictly fibrant G - I -simplicialsets. As the adjunction (1 . .
15) has a (canonical) simplicial enrichment, it thensuffices by Proposition A.2.8 and the characterization of the fibrant objects in G - I -SSet G -global provided by Theorem 1.3.38 that const sends fibrant objects tostatic ones, which is immediate from the definitions.In order to construct a canonical equivalence filling (1 . . . .
14) as (ev ω ) ∞ is an equivalence. (cid:3) In the world of G - I -simplicial sets there are rather explicit pointset models ofthe right adjoints R ev ∅ and R , which we will introduce now. To this end we define(1.3.16) U G := ∞ a i =0 a H ⊂ G finite G/H. If G is finite the above is just a particular construction of a complete G -setuniverse. However, in general this need not be countable and it can even haveuncountably many orbits (e.g. for G = L R Z / Z ). Despite these words of warningwe always have: Lemma . Let H be any finite group and let ι : H → G be any injectivegroup homomorphism. Then the H -set ι ∗ U G contains a complete H -set universe. Proof.
We may assume without loss of generality that ι is literally the inclu-sion of a finite subgroup H of G . We now have an H -equivariant injection ∞ a i =0 a K ⊂ H H/K → U G induced from H ֒ → G (and using that each subgroup K ⊂ H is in particular a finitesubgroup of G ). As the left hand side is a complete H -set universe, this finishesthe proof. (cid:3) By the universal property of enriched presheaves, ev U G has a simplicial rightadjoint given explicitly by R ( X )( A ) = maps( E Inj( A, U G ) , X )with the obvious functoriality in each variable and the conjugate G -action. We cannow prove: G -GLOBAL HOMOTOPY THEORY Proposition . The simplicial adjunction (1.3.17) u G := ev U G : G - I -SSet G -global ⇆ G -SSet proper : R is a Quillen adjunction with homotopical left adjoint. There are canonical equiva-lences (cid:0) ev U G (cid:1) ∞ ≃ R ev ∅ and R R ≃ R . Proof.
Let us use call a map of G - I -simplicial sets that becomes a E -weakequivalences after evaluating at ω an E -weak equivalence again. Because of thecanonical isomorphism ev U G ◦ const ∼ = id it then suffices that (1 . .
17) is a Quillenadjunction and that ev U G is homotopical in E -weak equivalences.If H ⊂ G is any finite subgroup, then we pick an injective homomorphism ι : H ′ → G with image H from a universal subgroup H ′ ⊂ M . With this notationwe then have for any G - I -simplicial set X an actual equality (cid:0) X ( U G ) (cid:1) H = X (cid:0) ι ∗ ( U G ) (cid:1) ι and analogously for morphisms. By the previous lemma there exists an H ′ -equi-variant embedding ω → ι ∗ ( U G ) (with respect to the tautological action on the lefthand side) and by Lemma 1.3.11 we conclude that the induced natural transforma-tion (–) ι ◦ ev ω ⇒ (–) H ◦ ev U G is a weak equivalence. Thus, ev U G is homotopical in E -weak equivalences (and hence in particular in G -global weak equivalences).It only remains to prove that ev U G sends the standard generating cofibrationsto proper cofibrations. As the proper model structure is simplicial, we are reducedto showing that the G -simplicial set I ( A, U G ) × ϕ G is cofibrant in the proper modelstructure (i.e. has finite isotropy groups) for any finite group H , any finite (faithful) H -set A and any group homomorphism ϕ : H → G . For this we let ( f , . . . , f n ; g )represent any n -simplex. If g ′ fixes [ f , . . . , f n ; g ], then in particular g ′ g = gϕ ( h )for some h ∈ H , and hence g ′ = gϕ ( h ) g − . As the right hand side can only takefinitely many values for any fixed g , the claim follows. (cid:3) Remark . For G = 1 the above adjunction is the I -analogue of [ Sch18 ,Remark 1.2.24 and Proposition 1.2.27]. As we show in Section 1.5, there exists azig-zag of homotopical functors I -SSet forget −−−→ I -SSet |·| −→ I -Top forget ←−−− L -Top where L -Top denotes Schwede’s orthogonal spaces (with respect to the globalmodel structure for the class of finite groups ), and all of these induce equivalenceson associated quasi-categories. It is then not hard to check directly that under thisidentification the adjunction const ∞ ⊣ (–) R E M ⊣ R corresponds to the adjunction L ⊣ R ev ⊣ R R considered in loc.cit. (for the trivial group).In fact, there is also a completely abstract way to prove this: namely, in bothcases the leftmost adjoint under consideration preserves the terminal object (in ourcase because we have explicitly constructed a further left adjoint, in Schwede’s caseby direct inspection). If we now have any equivalence Φ between L -Top ∞ and I -SSet ∞ , then both ways through the diagram SSet ∞ I -SSet ∞ L -Top ∞ const ∞ L Φ.4. TAMENESS 87 are cocontinuous and send the terminal object to the terminal object. By the univer-sal property of spaces there is therefore a contractible space of natural equivalencesfilling this.
In this section we will be concerned with the notion of tameness for M -actionsand E M -actions, and we will in particular show that the models from Section 1.2have tame analogues, that still model unstable G -global homotopy theory.The reason to study tameness here is twofold: firstly, the categories of tame M -and E M -simplicial sets carry interesting symmetric monoidal structures, which wewill study in Chapter 2, and which are related to the construction of global algebraic K -theory [ Sch19b ]; secondly, tame M - and E M -simplicial sets are intimatelyconnected to the diagram spaces considered in the previous section, and the theorydeveloped here will in particular allow us to give a complete characterization of the G -global weak equivalences in G - I -SSet (Theorem 1.4.27). M -actions. We begin with the notion of tame M -actions, which first appeared (for actions on abelian groups) as [ Sch08 , Defi-nition 1.5] and were then further studied (for actions on sets and simplicial sets)in [
SS20 ], see in particular [
SS20 , Definition 2.2 and Definition 3.1].
Definition . Let A ⊂ ω be any finite set. We write M A ⊂ M for thesubmonoid of those u ∈ M that fix A pointwise, i.e. u ( a ) = a for all a ∈ A .Let X be any M -set. An element x ∈ X is said to be supported on A if u.x = x for all u ∈ M A ; we write X [ A ] ⊂ X for the subset of those elements that aresupported on A , i.e. X [ A ] = X M A .We call x finitely supported if it is supported on some finite set and we write X τ := [ A ⊂ ω finite X [ A ] for the subset of all finitely supported elements. We call X tame if X = X τ .On M -simplicial sets everything can be extended levelwise: Definition . Let X be an M -simplicial set. An n -simplex x is supportedon A (for A ⊂ ω finite) if it is supported on A as an element of the M -set X n of n -simplices of X . Analogously, x is said to be finitely supported if it is so as anelement of X n . We define X τ via ( X τ ) n = ( X n ) τ , i.e. as the collection of all finitelysupported simplices. X is called tame if X τ = X .The finiteness of A is not necessary in the above definitions, but we will restrictto this case in the present article as this avoids some subtleties (that mostly appearwhen A has only finite complement in ω ).Let us record some basic properties of the above notions. All of these can befound explicitly stated in [ SS20 ] for M -sets and are easily extended to M -simplicialsets (for which they also appear implicitly in op. cit. ). Lemma . (1) If f : X → Y is a map of M -sets and A ⊂ ω is finite,then f restricts to f [ A ] : X [ A ] → Y [ A ] and hence in particular to f τ : X τ → Y τ . (2) If X is any M -simplicial set and A ⊂ ω finite, then X [ A ] is a simplicialsubset of X . In particular, X τ is a simplicial subset. G -GLOBAL HOMOTOPY THEORY (3) If f : X → Y is a map of M -simplicial sets and A ⊂ ω is finite, then f restricts to f [ A ] : X [ A ] → Y [ A ] . In particular, it restricts to f τ : X τ → Y τ . Proof.
The first statement is an easy calculation and we omit it; this also ap-pears without proof in [
SS20 , discussion before Lemma 2.6]. The second statementfollows by applying the first one to the structure maps, and the third one followsthen by applying the first one levelwise. (cid:3)
Lemma . Let u ∈ M and let X be any M -simplicial set. Then supp( u.x ) = u (supp( x )) for any finitely supported simplex x . In particular, the translation u. – : X → X restricts to X [ A ] → X [ u ( A )] for any finite A ⊂ ω , and X τ is a M -simplicial subset of X . Proof.
The first statement is [
SS20 , Proposition 2.5-(ii)], which immediatelyimplies the second one. The final statement then in turn follows from the secondone, also cf. [
SS20 , discussion after Proposition 2.5-(ii)]. (cid:3)
Lemma . Let X ∈ M -SSet , u, u ′ ∈ M , and let x ∈ X n . Assume that X is supported on some finite set A ⊂ ω such that u | A = u ′ | A . Then u.x = u ′ .x . Proof.
This is [
SS20 , Proposition 2.5-(i)], applied to the M -set X n . (cid:3) Definition . Let X be any M -set and let x ∈ X . Then the support supp( x ) is the intersection of all finite sets A ⊂ ω on which x is supported. Inparticular, supp( x ) = ω if x is not finitely supported. Lemma . In the above situation, x is supported on supp( x ) . Proof.
This is immediate from [
SS20 , Proposition 2.3]. (cid:3)
The structure of tame M - G -simplicial sets. [ SS20 , Theorem 2.11],which we recall below, describes how tame M -sets decompose into some standardpieces. As a consequence of this we will prove below: Theorem . Let us define (1.4.1) I tame := { (Inj( A, ω ) × Σ A X ) × ∂ ∆ n ֒ → (Inj( A, ω ) × Σ A X ) × ∆ n : A ⊂ ω finite, X any G - Σ A -biset, n ≥ } . Then the I tame -cell complexes are precisely the tame M - G -simplicial sets. If X is any M -set, let us write s n ( X ) for the subset of those x ∈ X withsupp( x ) = { , . . . , n } . Lemma 1.4.5 provides for any u ∈ Inj(
A, ω ) a well-definedmap s n ( X ) → X obtained by acting with any extension of u to an ¯ u ∈ M , and[ SS20 , Proposition 2.5-(ii)] shows that this restricts to a Σ n -action on s n ( X ). Theorem . Let X be any tame M -set. Then the map (1.4.2) ∞ a n =0 Inj( { , . . . , n } , ω ) × Σ n s n ( X ) → X induced by the above construction is well-defined and an isomorphism of M -sets. Proof.
This is [
SS20 , Theorem 2.11]. (cid:3)
Corollary . Let X be any tame M - G -set. Then each s n ( X ) is a G -subset of X , and with respect to this action the map (1 . . is an isomorphism of M - G -sets. .4. TAMENESS 89 Proof.
In order to prove that s n ( X ) is closed under the action of G , we haveto show that supp( g.x ) = supp( x ) for all g ∈ G and x ∈ X . The inclusion ‘ ⊂ ’ is aninstance of Lemma 1.4.3 because the G -action commutes with the M -action. Theinclusion ‘ ⊃ ’ then follows by applying the same argument to g − and g.x .Again using that the M -action commutes with the G -action, we see that (1 . . G -equivariant, hence an isomorphism of G - M -sets by the previous theorem. (cid:3) The only remaining ingredient for the proof of Theorem 1.4.8 is the following:
Lemma . Let X be a tame M - G -set and let Y ⊂ X be a G - M -subset.Then also X r Y is a M - G -subset. We caution the reader that the above is not true in general for non-tameactions—for example, the subset Y ⊂ M of non-surjective maps is closed underthe M -action, but its complement is not. Proof.
The set X r Y is obviously closed under the G -action. The fact thatit is moreover closed under the M -action appeared in a previous version of [ SS20 ];let us give the argument for completeness. Assume x ∈ X r Y and let u ∈ M suchthat u.x ∈ Y . By tameness, there exists a finite set A on which x is supported. Wenow pick any invertible v ∈ M such that v | A = u | A . By Lemma 1.4.5 we then have v.x = u.x ∈ Y and hence also x = v − . ( v.x ) ∈ Y , which is a contradiction. (cid:3) Proof of Theorem 1.4.8.
Obviously all sources and targets of maps in I tame are tame. As the tame M - G -simplicial sets are closed under all colimits, we seethat all I tame -cell complexes are tame (cf. Lemma 1.1.7).Conversely, let X be any tame M - G -simplicial set; we consider the usual skele-ton filtration ∅ = X ( − ⊂ X (0) ⊂ X (1) ⊂ · · · of X . It suffices to prove that each X ( n − → X ( n ) is a relative I tame -cell complex.For this we contemplate the (a priori non-equivariant) pushout X nondeg n × ∂ ∆ n X nondeg n × ∆ n X ( n − X ( n ) where X nondeg n denotes the subset of nondegenerate n -simplices. The degenerate simplices obviously form an M - G -subset of X n and hence so do the nondegenerateones by the previous lemma. With respect to this action, all maps in the abovesquare are obviously M - G -equivariant so that this is a pushout in M - G -SSet .But applying Corollary 1.4.10 to X nondeg n expresses the top horizontal map as acoproduct of maps in I tame , finishing the proof. (cid:3) Connection to I -simplicial sets. Sagave and Schwede showed that tame M -simplicial sets are equivalent as a 1-category to the full subcategory of I -simplicial sets spanned by the flat objects, i.e. those that are cofibrant in the globalmodel structure. In order to state their precise comparison, we need: Construction . Let X be any M -simplicial set. We write X • for the I -simplicial set with ( X • )( A ) = X [ A ] ; an injection j : A → B acts by extendingit to an injection ¯ j ∈ M and then using the M -action (this is well-defined byLemma 1.4.4 together with Lemma 1.4.5). This becomes a functor by functoriality G -GLOBAL HOMOTOPY THEORY of the individual X [ A ] ; in particular, if G is any group we get an induced functor(–) • : M - G -SSet → G - I -SSet .The inclusions X [ A ] ֒ → X assemble into a natural map ǫ : X • ( ω ) → X for any M - G -simplicial set X . Moreover, if Y is a G - I -simplicial set, then the structuremap Y ( A ) → Y ( ω ) for A ⊂ ω factors through Y ( ω ) [ A ] , and for varying A theseassemble into a natural map Y → Y ( ω ) • .The above construction is a ‘coordinate free’ version of [ SS20 , Construction 5.5]applied in each simplicial degree (and with G -actions pulled through), also cf. [ SS20 ,discussion before Corollary 5.7]. In particular, [
SS20 , Proposition 5.6] implies, alsocf. [
SS20 , Corollary 5.7]:
Lemma . The above defines a simplicial adjunction (1.4.3) ev ω : G - I -SSet ⇄ M - G -SSet : (–) • where the left adjoint has image in M - G -SSet τ , and for any M - G -simplicial set X the counit X • ( ω ) → X factors through an isomorphism onto X τ .Moreover, the right adjoint has essential image the flat G - I -simplicial sets,i.e. η : Y → Y ( ω ) • is an isomorphism if and only if Y is flat. (cid:3) E M -actions. We now want to introduce and study analoguesof tameness and support for E M -simplicial sets. Definition . Let A ⊂ ω be finite and let X be an E M -simplicial set.An n -simplex x of X is said to be supported on A ⊂ ω if E ( M A ) acts trivially on x , i.e. the composition E ( M A ) × ∆ n ֒ → E M × ∆ n E M× x −−−−−→ E M × X act −−→ X agrees with E ( M A ) × ∆ n pr −→ ∆ n x −→ X. The simplex x is said to be finitely supported if it is supported on some finite set A .We write X τ = S A ⊂ ω finite X [ A ] for the collection of all finitely supported simplices,and we call X tame if X = X τ , i.e. if all its simplices are finitely supported.1.4.2.1. Comparison of supports.
It is a straight-forward but somewhat lengthyendeavor to verify the analogues of Lemmas 1.4.3–1.4.5 above in the world of E M -simplicial sets. We will not do this at this point as they have shorter proofs oncewe know the following theorem, that is also of independent interest: Theorem . Let X be an E M -simplicial set, let n ≥ , and let A be anyfinite set. Then an n -simplex x is supported on A in the sense of Definition 1.4.14if and only if it is supported on A as a simplex of the underlying M -simplicial setof X (see Definition 1.4.2). In other words, X [ A ] = (forget X ) [ A ] as simplicial sets,where forget denotes the forgetful functor E M -SSet → M -SSet .In particular, X [ A ] is a simplicial subset of X and this yields a subfunctor (–) [ A ] : E M -SSet → SSet of the forgetful functor. This functor is corepresentedin the enriched sense by E Inj(
A, ω ) via evaluation at the -simplex given by theinclusion A ֒ → ω . The proof requires some combinatorial preparations:
Proposition . Let A ⊂ ω be any finite set, and let u , . . . , u n ∈ M .Then there exists a χ ∈ M A such that im( u χ ) ∪ · · · ∪ im( u n χ ) has infinite comple-ment in ω . .4. TAMENESS 91 Proof.
We will construct strictly increasing chains B ( B ( · · · and C ( C ( · · · of finite subsets of ω r A such that for all j ≥ C j ∩ n [ i =0 u i ( B j ) = ∅ . Let us first show how this yields the proof of the claim: we set B ∞ := S ∞ j =0 B j and C ∞ := S ∞ j =0 C j . Then both of these are infinite sets and moreover each u i ( B ∞ )misses C ∞ by Condition (1 . .
4) and since the unions are increasing. As B ∞ isinfinite, we find an injection c : ω r A → ω with image B ∞ . We claim that χ := c + id A has the desired properties: indeed, this is again an injection as B ∞ ∩ A = ∅ ,and it is the identity on A by construction. On the other hand, n [ i =0 im( u i χ ) = n [ i =0 u i ( A ) | {z } =: A ′ ∪ n [ i =0 u i ( B ∞ ) | {z } =: B ′ and B ′ has infinite complement in ω (namely, at least C ∞ ) whereas A ′ is even finite;we conclude that also their union has infinite complement as desired.It therefore only remains to construct the chains B ( · · · and C ( · · · forwhich we will proceed by induction. We begin by setting B = C = ∅ whichobviously has all the required properties. Now assume we’ve already constructedthe finite sets B j and C j satisfying (1 . . (cid:0) A ∪ B j ∪ S ni =0 u − i ( C j ) (cid:1) is finite as A, B j , C j are finite and each u i isinjective, so we can pick a b ∈ ω not contained in it. We now set B j +1 := B j ∪ { b } ,which is obviously finite and a proper superset of B j by construction. We moreoverobserve that C j misses all u i ( B j +1 ) as it misses u i ( B j ) by the induction hypothesisand moreover u i ( b ) / ∈ C j by construction.By the same argument we can pick c ∈ ω r ( A ∪ C j ∪ S ni =0 u i ( B j +1 )) anddefine C j +1 := C j ∪ { c } ; this is obviously again finite and a proper superset of C j .We claim that Condition (1 . .
4) is satisfied for B j +1 and C j +1 . Indeed, we havealready seen that C j misses all of u i ( B j +1 ). On the other hand, also c / ∈ u i ( B j +1 )by construction, verifying the condition. This finishes the inductive constructionsand hence the proof of the proposition. (cid:3) Proposition . Let A ⊂ ω be finite, and let ( u , . . . , u n ) , ( v , . . . , v n ) ∈M n such that u i | A = v i | A for i = 0 , . . . , n . Then [ u , . . . , u n ] = [ v , . . . , v n ] in M n / M A . Proof.
Applying the above to the 2 n + 2 injections u , . . . , u n , v , . . . , v n , wemay assume without loss of generality that B := ω r n [ i =0 im u i ∪ n [ i =0 im v i ! is infinite. We can therefore choose an injection ϕ : ω r A → ω with image in B ,and we moreover pick a bijection ω r A ∼ = ( ω r A ) ∐ ( ω r A ), yielding two injections j , j : ω r A → ω r A whose images partition ω r A . We now define for i = 0 , . . . , nw i ( x ) := u i ( x ) if x ∈ Au i ( y ) if x = j ( y ) for some y ∈ ω r Aϕ ( y ) if x = j ( y ) for some y ∈ ω r A . G -GLOBAL HOMOTOPY THEORY This is indeed well-defined as ω is the disjoint union A ⊔ im( j ) ⊔ im( j ) and since j and j are injective. We now claim that w i is injective (and hence an element of M ):indeed, assume w i ( x ) = w i ( x ′ ) for x = x ′ . Since im( u i ) is disjoint from im ϕ andsince ϕ is injective, we conclude that x, x ′ / ∈ im j . As moreover w i | A = u i | A and w i j = u i | ω r A are injective, we can assume without loss of generality that x ∈ A and x ′ = j ( y ′ ) for some y ′ ∈ ω r A . But then u i ( x ) = w i ( x ) = w i ( x ′ ) = u i ( y ′ ),which contradicts the injectivity of u i as y ′ / ∈ A and hence in particular y ′ = x .We now observe that by construction w i (incl A + j ) = u i and w i (incl A + j ) = u i | A + ϕ . On the other hand, incl A + j and incl A + j are obviously injectionsfixing A pointwise, so that they witness the equalities[ u , . . . , u n ] = [ w , . . . , w n ] = [ u | A + ϕ, . . . , u n | A + ϕ ]in M n +1 / M A . Analogously, one shows that [ v , . . . , v n ] = [ v | A + ϕ, . . . , v n | A + ϕ ],and since v i | A = u i | A by assumption, this further agrees with [ u | A + ϕ, . . . , u n | A + ϕ ], finishing the proof. (cid:3) Corollary . Let ( u , . . . , u n ) , ( u ′ , . . . , u ′ n ) ∈ ( E M ) n , let X be an E M -simplicial set, and let x ∈ X n . Assume that x is supported as an element of the M -set X n on some finite set A such that u i | A = u ′ i | A for i = 0 , . . . , n . Then ( u , . . . , u n ) .x = ( u ′ , . . . , u ′ n ) .x. Proof.
We begin with the special case that there exists α ∈ M A such that u ′ i = u i α for all i . In this case( u ′ , . . . , u ′ n ) .x = ( u α, . . . , u n α ) .x = ( u , . . . , u n ) .α.x = ( u , . . . , u n ) .x as desired, where the last step uses that x is fixed by α ∈ M A .We conclude that M n → X n , ( u , . . . , u n ) ( u , . . . , u n ) .x descends to M n / M A ; the claim follows therefore from the previous proposition. (cid:3) Proof of Theorem 1.4.15.
Let χ : E Inj(
A, ω ) × ∆ n → X be any E M -equivariant map. We claim that the image x of the n -simplex ( i, . . . , i ; id [ n ] ), where i denotes the inclusion A ֒ → ω , is supported on A . Indeed, let f : [ m ] → [ n ] be anymap in ∆ and let ( u , . . . , u m ) ∈ M m +1 A . Then( u , . . . , u m ) .f ∗ x = ( u , . . . , u m ) χ ( i, . . . , i ; f ) = χ ( u i, . . . , u m i ; f )= χ ( i, . . . , i ; f ) = f ∗ x where we have used in this order: the definition of x and that χ is simplicial; that χ is E M -equivariant; that ui = i for all u ∈ M A ; that χ is simplicial; the definitionof x and that χ is simplicial. This proves the claim.On the other hand, let x ∈ X n be supported on A with respect to the under-lying M -action . We define χ m : ( E Inj(
A, ω ) × ∆ n ) m → X m as follows: we senda tuple ( u , . . . , u m ; f ), where the u i are injections A → ω and f : [ m ] → [ n ] isany map in ∆, to ( e u , . . . , e u m ) .f ∗ x , where each e u i is an extension of u i to all of ω , i.e. to an element of M . Such extensions can certainly be chosen as A is finite,and we claim that this is in fact independent of the choice of extension: indeed,as x is fixed by M A , so is f ∗ x , and hence this follows from the previous corollary.With this established it is trivial to prove that the χ m assemble into a simplicialmap E Inj(
A, ω ) × ∆ n → X and that this is E M -equivariant. Moreover, a pos-sible extension of i : A ֒ → ω is given by the identity of ω and hence we see that χ m ( i, . . . , i ; id [ n ] ) = x . .4. TAMENESS 93 We conclude that if x ∈ X n is supported on A with respect to the underlying M -action, then it is obtained by evaluating some E M -equviariant χ : E Inj(
A, ω ) × ∆ n → X at the canonical element (by the second paragraph), and hence it isactually supported on A with respect to the E M -action (by the first one). Asthe converse holds for trivial reasons, we conclude that the two notions of ‘beingsupported on A ’ indeed agree. It then follows from the above that evaluation atthe canonical element defines a surjection(1.4.5) maps( E Inj(
A, ω ) , X ) → X [ A ] . We know from Lemma 1.4.3 that X [ A ] (which we may now view as definedwith respect to the M -action) is a subfunctor of the forgetful functor E M -SSet → SSet , and with respect to this (1 . .
5) is obviously natural. To finish the proof of theclaimed corepresentability result, it therefore suffices that (1 . .
5) is also injective.For this we let χ, χ ′ : E Inj(
A, ω ) × ∆ n → X with χ ( i, . . . , i ; id [ n ] ) = χ ′ ( i, . . . , i ; id [ n ] ).Then we have for any ( u , . . . , u m ) ∈ M m +1 and f : [ m ] → [ n ] in ∆ χ ( u , . . . , u m ; f ) = ( e u , . . . , e u m ) .χ ( i, . . . , i ; f ) = ( e u , . . . , e u m ) .f ∗ χ ( i, . . . , i ; id [ n ] )= ( e u , . . . , e u m ) .f ∗ χ ′ ( i, . . . , i ; id [ n ] ) = χ ′ ( u , . . . , u m ; f )where again e u i is any extension of u i to an element of M . This finishes the proofof corepresentability and hence of the theorem. (cid:3) We will later need the following computation following from the above proof:
Corollary . For any simplicial set K , any finite set A and any G - Σ A -biset X , the map E M × M (cid:0) (Inj( A, ω ) × Σ A X ) × K (cid:1) → ( E Inj(
A, ω ) × Σ A X ) × K adjunct to the product of the inclusion of the -simplices with the identity of K , isan isomorphism of E M - G -simplicial sets. Proof. As E M × M – is a simplicial left adjoint, we are reduced to provingthat the corresponding map E M × M Inj(
A, ω ) → E Inj(
A, ω ) is an isomorphism (itis obviously left- M -right-Σ A -equivariant). For this we observe that by the abovetheorem E Inj(
A, ω ) corepresents (–) [ A ] by evaluating at the inclusion ι : A ֒ → ω . Onthe other hand, it is obvious that Inj( A, ω ) corepresents (–) [ A ] : M -SSet → SSet by evaluating at the same element, also see [
SS20 , Example 2.9]. By adjointness, E M × M Inj(
A, ω ) therefore corepresents (–) [ A ] ◦ forget via evaluating at [1; ι ]. Bythe previous theorem this agrees with (–) [ A ] , and as the above map sends [1; ι ] to ι , this completes the proof. (cid:3) It is now immediate that Lemma 1.4.3 holds verbatim with M replaced by E M everywhere (and part of this already appears in the statement of the abovetheorem). The analogues of the remaining lemmas are a bit more interesting: forLemma 1.4.5 this was done in Corollary 1.4.18, so it only remains to consider theanalogue of Lemma 1.4.4: Lemma . Let ( u , . . . , u n ) ∈ ( E M ) n , let A ⊂ ω be finite and let X beany E M -simplicial set. Then the composition (1.4.6) ∆ n × X [ A ] ( u ,...,u n ) × incl −−−−−−−−−−→ E M × X act −−→ X has image in X [ u ( A ) ∪···∪ u n ( A )] ; in particular, X τ is an E M -simplicial subset of X . G -GLOBAL HOMOTOPY THEORY Proof.
It suffices to prove the first statement. For this we observe that anysimplex in the image of (1 . .
6) can be written as (cid:0) f ∗ ( u , . . . , u n ) (cid:1) .x for some m -simplex x of X [ A ] and some f : [ m ] → [ n ] in ∆. We now calculate for any v ∈ M (1.4.7) v. ( f ∗ ( u , . . . , u n ) .x ) = v. ( u f (0) , . . . , u f ( m ) ) .x = ( vu f (0) , . . . , vu f ( n ) ) .x. If now v is the identity on u ( A ) ∪ · · · ∪ u n ( A ), then vu f ( i ) and u f ( i ) agree on A forall i = 0 , . . . , m . Hence, if x is supported on A , then( vu f (0) , . . . , vu f ( m ) ) .x = ( u f (0) , . . . , u f ( m ) ) .x = f ∗ ( u , . . . , u n ) .x by Corollary 1.4.18. Together with (1 . .
7) this precisely yields the claim. (cid:3)
Remark . Of course, we could have just turned Theorem 1.4.15 into thedefinition instead—however, this does not buy as anything, as we would have theninstead to do all of the above work in order to prove Corollary 1.4.18 (or, whatis in this case equivalent, the corepresentability statement), which will be crucialbelow. Moreover, defining support by means of the mere M -action is ‘evil’ from ahomotopical viewpoint, as we a priori throw away a lot of higher information: forexample, if u ∈ M and x is a vertex of an E M -simplicial set, then we can think ofthe edge ( u, . ( s ∗ x ), where s : [1] → [0] is the unique morphism in ∆, as providing anatural comparison between x and u.x . From the homotopical viewpoint we shouldthen always be interested in this edge itself and not only in its endpoints.We can now use the above to prove the following comparison result: Theorem . The simplicial adjunction E M × M (–) ⊣ forget restricts to (1.4.8) E M × M – : M - G -SSet τ ⇄ E M - G -SSet τ : forget . Both functors in (1 . . preserve and reflect G -global weak equivalences, and theydescend to mutually inverse equivalences on associated quasi-categories. Proof.
The forgetful functor obviously restricts to the full subcategories oftame objects. To see that also E M × M – restricts accordingly, we appeal to The-orem 1.4.8: As the tame E M - G -simplicial sets are closed under all colimits, weare reduced (Lemma 1.1.7) to showing that E M × M (cid:0) (Inj( A, ω ) × Σ A X ) × K (cid:1) istame for every G -Σ A -biset X and any simplicial set K , which is immediate fromCorollary 1.4.19.The forgetful functor creates weak equivalences (even without the tameness as-sumption) as it is homotopical and part of a Quillen equivalence by Corollary 1.2.40.Let us now prove that the unit η : Y → forget E M × M Y is a G -semistableweak equivalence for any Y ∈ M - G -SSet τ . We caution the reader that this is nota formal consequence of Corollary 1.2.40 because we are not deriving E M × M –in any way here. Instead, we need to use Theorem 1.2.49 together with the fullgenerality of Theorem 1.2.56:Both E M × M – as well as forget are left adjoints and hence cocontinuous. Asboth functors preserve tensors, the composition forget( E M × M –) sends the mapsin I tame to underlying cofibrations. Corollary 1.1.8 therefore reduces this to the casethat Y = (Inj( A, ω ) × Σ A X ) × K for H, A, X as above and K any simplicial set.Again using that both E M × M – and forget preserve tensors (and that the aboveis a simplicial adjunction) we reduce further to the case that Y = Inj( A, ω ) × Σ A X . .4. TAMENESS 95 After postcomposing with the isomorphism from Corollary 1.4.19, the unit simplybecomes the inclusion of the 0-simplices and this factors asInj(
A, ω ) × Σ A X π −→ (cid:0) Inj(
A, ω ) × Σ A X (cid:1) // M → E Inj(
A, ω ) × Σ A X, where the right hand map is the G -universal weak equivalence from Theorem 1.2.56.As the left hand map is moreover a G -global weak equivalence by Theorem 1.2.49,thus so is the unit.Let now f : X → Y be any map in M - G -SSet τ . In the naturality square X Y forget E M × M X forget E M × M Y η f η forget E M× M f both vertical maps are G -global weak equivalences by the above. Thus, if f is a G -global weak equivalence, then so is the lower horizontal map. But forget reflectsthose, hence also E M× M f is a G -global weak equivalence, proving that E M× M –is homotopical. Conversely, if E M × M f is a G -global weak equivalence, then so isforget E M × M f and hence also f by the above square, i.e. E M × M – also reflectsweak equivalences.Finally, if X ∈ E M - G -SSet τ is arbitrary, then the triangle identity for adjunc-tions shows that forget ǫ X : forget E M × M (forget X ) → forget X is right inverse to η forget X , hence a G -global weak equivqalence. As forget reflects these, we concludethat also ǫ is levelwise a G -global weak equivalence, proving that the functors in(1 . .
8) induce mutually inverse equivalences of quasi-categories. (cid:3)
Connection to I -simplicial sets. We can now use Theorem 1.4.15 togive another model of the equivalence between the homotopy theories of G - I -SSet and E M - G -SSet in the spirit of Lemma 1.4.13. Construction . Let X be any E M -simplicial set. We write X • for the I -simplicial set with ( X • )( A ) = X [ A ] ; an ( n + 1)-tuple of injections j , . . . , j n : A → B acts by extending each of them to an injection ¯ j k : ω → ω and then using the E M -action (this is well-defined by Corollary 1.4.18 together with Lemma 1.4.20).This becomes a functor by functoriality of the individual X [ A ] ; in particular, if G is any group we get an induced functor (–) • : E M - G -SSet → G - I -SSet .We define ǫ : X • ( ω ) → X as the map induced by the inclusions X [ A ] ֒ → X .Moreover, if Y ∈ G - I -SSet , then Y ( A ) → Y ( ω ) factors through Y ( ω ) [ A ] by defi-nition of the action, and one easily checks that these assemble into η : Y → Y ( ω ) • . Remark . By Theorem 1.4.15, the diagram(1.4.9) G - E M -SSet τ G - I -SSet G - M -SSet τ G - I -SSet (–) • forget forget(–) • commutes strictly, and the same can be arranged for ev ω by construction. Un-der these identifications, also the unit and counit are preserved, i.e. forget( η Y ) = η forget Y and forget( ǫ X ) = ǫ forget X .As the notions of tameness in E M - G -SSet and M - G -SSet agree by anotherapplication of Theorem 1.4.15, we conclude from Lemma 1.4.13: G -GLOBAL HOMOTOPY THEORY Lemma . The above yields a simplicially enriched adjunction (1.4.10) ev ω : G - I -SSet ⇄ E M - G -SSet : (–) • where the left adjoint has image in E M - G -SSet τ . Moreover, for any E M - G -simplicial set X the counit ( X • )( ω ) → X factors through an isomorphism onto X τ ,and the essential image of (–) • consists precisely of those G - I -simplicial sets thatare flat, i.e. whose underlying I -simplicial sets are globally cofibrant. (cid:3) Theorem . The adjunction (1 . . defines a simplicial Quillen equiva-lence with respect to the G -global injective model structures. Proof.
It is clear that ev ω preserves G -global weak equivalences and injectivecofibrations, so (1 . .
10) is a Quillen adjunction.To see that it is a Quillen equivalence, we observe that, since ev ω is homotopical,its left derived functor agrees with (ev ω ) ∞ which was seen to be an equivalence inTheorem 1.3.30. (cid:3) G -global weak equivalences of G - I -simplicial sets. We can nowfinally give an alternative description of the weak equivalences in G - I -SSet : Theorem . The following are equivalent for a map f in G - I -SSet : (1) f is a G -global weak equivalence in G - I -SSet . (2) I × I f is a G -global weak equivalence in G - I -SSet . (3) f ( ω ) is a G -global weak equivalence in M - G -SSet . (4) E M × M f ( ω ) is a G -global weak equivalence in E M - G -SSet . We emphasize that the functors in (2) and (4) are not derived in any way.
Proof.
The total mate of (1 . .
9) provides a natural isomorphism filling E M - G -SSet τ G - I -SSet M - G -SSet τ G - I -SSet . ev ω E M× M – ev ω I× I – With this established, (2) ⇔ (4) follows immediately from the definitions.On the other hand, Theorem 1.4.22 in particular tells us that (3) ⇔ (4) andthat E M× M – is homotopical when restricted to M - G -SSet τ . It therefore followsfrom the above isomorphism that I × I – sends weak equivalences at infinity to G -global weak equivalences. In particular, I × I – : G - I -SSet restricted → G - I -SSet (restricted) global is fully homotopical, after which (1) ⇔ (2) follows again from the definitions. (cid:3) We can now compare the adjunction (1 . .
3) to (1 . . Corollary . There exists a canonical equivalence of quasi-functors L l ≃ (ev ω ) ∞ : G - I -SSet ∞ restricted G -global → M - G -SSet ∞ G -global . In particular, (ev ω ) ∞ is an equivalence of quasi-categories. Proof.
The previous theorem guarantees that ev ω is homotopical and henceindeed descends to such a quasi-functor. With this established, Proposition 1.3.28exhibits it as left quasi-inverse to R (–)[ ω • ]. The claim follows because L l is quasi-inverse to R (–)[ ω • ] by Theorem 1.3.31. (cid:3) .4. TAMENESS 97 Further model structures.
With the above characterization at hand, wecan now introduce G -global model structures on G - I -SSet based on the strict andpositive model structures: Theorem . There is a unique model structure on G - I -SSet whose cofi-brations are the strict cofibrations and whose weak equivalences are the G -globalweak equivalences. This model structure is left proper, combinatorial, simplicial,and filtered colimits in it are homotopical. Moreover, its fibrant objects are pre-cisely the static strictly fibrant ones.Finally, the simplicial adjunctions I × I – : G - I -SSet G -global ⇄ G - I -SSet G -global : forgetid : G - I -SSet restricted G -global ⇄ G - I -SSet G -global : id are Quillen equivalences. Proof.
Let X be any G - I -simplicial set. We define X ∞ ( A ) = X (cid:0) ( A + ) ∞ (cid:1) together with the obvious structure maps. We have seen in the proof of Theo-rem 1.3.38 that ( A + ) ∞ has a free H -orbit for any faithful H -set A ; we conclude bythe usual argument that X ∞ is static whenever X is restricted static.The diagonal maps A → ( A + ) ∞ assemble into a map ι : X → X ∞ of G - I -simplicial sets. If X is restricted static, then this is a restricted weak equivalence:indeed, if A is any finite H -set, then ι factors in degree A as X ( A ) → colim A ⊂ B ⊂ ( A + ) ∞ finite H -subset X ( B ) ∼ = X (cid:0) ( A + ) ∞ (cid:1) where the left hand arrow is the structure map of the colimit and the isomorphismcomes from the usual cofinality argument. If A contains a free H -orbit, then theleft hand arrow is a G H,G -weak equivalence, as all transition maps in the (filtered!)colimit system are, and hence so is the composition.Similarly, one shows that if X is static, then ι is a strict weak equivalence. Theassignment X X ∞ becomes a functor in the obvious way, and one concludesfrom the above that this is homotopy inverse to the inclusion(static G - I -simplicial sets) strict w.e. ֒ → (restr. static G - I -simplicial sets) restr. w.e. so that the latter in particular induces an equivalence on quasi-localizations. Sim-ilarly, one shows that also(static G - I -simplicial sets) strict w.e. ֒ → (restr. static G - I -simplicial sets) restr. w.e. induces an equivalence on quasi-localizations (or one instead simply observes thatthe inclusion of both source and target to G - I -SSet global do).With this established, an easy diagram chase shows that the right derived func-tor of forget: G - I -SSet global → G - I -SSet strict is fully faithful with essential imageprecisely the static G - I -simplicial sets. Thus, all of the above claims follow fromTheorem A.2.5, modulo the following caveat already mentioned in Warning 1.3.42:the weak equivalences of the model structure obtained this way are defined in termsof I × L strict I – (i.e. where we derive with respect to the strict model structure), whilethe G -global weak equivalences are defined in terms of I × L restricted I – (i.e. where wederive with respect to the restricted model structure). However, we have seen inTheorem 1.4.27 that I × I – is fully homotopical with respect to the restricted (andhence also in the strict) weak equivalences in the source and the G -global ones inthe target, so both of these functors actually agree. (cid:3) G -GLOBAL HOMOTOPY THEORY One now concludes just as in Corollary 1.3.51:
Corollary . The simplicial adjunction forget : G - I -SSet G -global ⇄ G - I -SSet G -global : maps I ( I , –) is a Quillen adjunction with respect to the G -global model structures. (cid:3) Similarly to the above theorem one proves:
Theorem . There is a unique model structure on G - I -SSet whose cofi-brations are the positive cofibrations and whose weak equivalences are the G -globalweak equivalences. This model structure is left proper, combinatorial, simplicial,and filtered colimits in it are homotopical. Moreover, its fibrant objects are pre-cisely the static strictly fibrant ones. Finally, the simplicial adjunctions I × I – : G - I -SSet positive G -global ⇄ G - I -SSet positive G -global : forgetid : G - I -SSet positive G -global ⇄ G - I -SSet G -global : id are Quillen equivalences. (cid:3) Functoriality for G - I -simplicial sets. With the explicit description ofthe G -global weak equivalences at hand, most of the results about G - I -SSet trans-fer easily to G - I -SSet . We will demonstrate this for the functoriality propertieshere: Corollary . Let α : H → G be any group homomorphism. Then α ! : H - I -SSet H -global ⇄ G - I -SSet G -global : α ∗ is a simplicial Quillen adjunction with fully homotopical right adjoint, and likewisefor the corresponding restricted or positive model structures on either side. Proof.
One proves as in Lemma 1.3.52 that these are Quillen adjunctions. Toprove that α ∗ is homotopical, we consider the commutative diagram G - I -SSet G - I -SSet H - I -SSet H - I -SSet . I× I – α ∗ α ∗ I× I – The horizontal arrows preserve and reflect weak equivalences by Theorem 1.4.27while the right hand vertical arrow is homotopical by Lemma 1.3.52; the claimfollows immediately. (cid:3)
Corollary . Let α : H → G be any group homomorphism. Then α ∗ : G - I -SSet G -global injective ⇄ H - I -SSet H -global : α ∗ is a simplicial Quillen adjunction. Proof.
By the previous corollary α ∗ is homotopical and it obviously preservesinjective cofibrations, so it is left Quillen. (cid:3) Corollary . Let α : H → G be an injective group homomorphism. Then α ! : H - I -SSet H -global injective ⇄ G - I -SSet G -global injective : α ∗ is a simplicial Quillen adjunction. In particular, α ! is homotopical. .4. TAMENESS 99 Proof.
One proves analogously to Lemma 1.3.54 that α ! preserves injectivecofibrations. We now consider the commutative square on the left in H - I -SSet H - I -SSet G - I -SSet G - I -SSet forget α ∗ forget α ∗ H - I -SSet H - I -SSet G - I -SSet G - I -SSet . α ! I× I – α ! I× I – Passing to total mates yields a canonical isomorphism filling the square on theright. But the horizontal arrows in this preserve and reflect weak equivalences byTheorem 1.4.27 while the right hand vertical arrow is homotopical by Lemma 1.3.54.The claim follows immediately. (cid:3)
Remark . As before one concludes that the above change of group func-tors are compatible with all the equivalences we have constructed between thedifferent models of unstable G -global homotopy theory. Using the above results, we can nowprove that already the tame E M - G -simplicial sets (or M - G -simplicial sets) sufficeto model unstable G -global homotopy theory. Theorem . There exists a unique model structure on E M - G -SSet τ inwhich a map f is a weak equivalence, fibration, or cofibration, if and only if f • isa weak equivalence, fibration, or cofibration, respectively, in the positive G -globalmodel structure on G - I -SSet . We call this the positive G -global model structure .Its weak equivalences are precisely the G -global weak equivalences, and hence theyare in particular stable under filtered colimits.The model structure is proper, simplicial, and combinatorial with generatingcofibrations (1.4.11) ( E Inj(
A, ω ) × ϕ G ) × ∂ ∆ n ֒ → ( E Inj(
A, ω ) × ϕ G ) × ∆ n where H runs through finite groups, A = ∅ is a finite faithful H -set, ϕ is a homo-morphism H → G , and n ≥ .Finally, the simplicial adjunctions (1.4.12) incl : G - E M -SSet τ ⇄ G - E M -SSet injective G -global : (–) τ and (1.4.13) ev ω : G - I -SSet ⇄ G - E M -SSet τ : (–) • are Quillen equivalences. While this may sound like another application of our transfer criterion, it isin this case actually easier to verify the model structure axioms directly, as theadjunction ev ω ⊣ (–) • is already well-behaved 1-categorically, also see [ SS20 , The-orem 5.10 and Corollary 5.11] for an analogous argument in the non-equivariantsituation:
Proof.
The adjunction ev ω ⊣ (–) • exhibits E M - G -SSet τ as accessible Bous-field localization of the locally presentable category G - I -SSet , so it is itself locallypresentable.Let us now show that the above defines a model structure on E M - G -SSet τ .The 2-out-of-3 property for weak equivalences as well as the closure under retracts
00 1. UNSTABLE G -GLOBAL HOMOTOPY THEORY for all three classes are obvious. Moreover, as (–) • is fully faithful, the lifting axiomsare inherited from the lifting axioms for the positive G -global model structure.It only remains to verify the factorization axioms, for which we let f : X → Y be any map of tame E M - G -simplicial sets. Then we can factor f • as a cofibration i : X • → Z followed by an acyclic fibration p : Z → Y • . But cofibrations in thepositive G -global model structure are in particular cofibrations in I -SSet (Propo-sition 1.3.49 together with Lemma 1.3.55), so Z is flat again and hence lies in theessential image of (–) • . We can therefore assume without loss of generality that Z = Z ′• for some Z ∈ E M - G -SSet τ . By full faithfulness of (–) • we can thenwrite i = i ′• , p = p ′• , which yields the desired factorization f = p ′ i ′ . The remainingfactorization axiom is proven analogously.This completes the proof of the existence of the positive G -global model struc-ture. By definition, f : X → Y is a weak equivalence if and only if f • is, which inturn is equivalent by definition to f • ( ω ) being a G -global weak equivalence of E M - G -simplicial sets. As (–) • is fully faithful, f • ( ω ) is conjugate to f , which shows thatthe weak equivalences are precisely the G -global weak equivalences. In particular,they are stable under filtered colimits.The model structure is right proper since it is transferred from a right propermodel stucture. Moreover, E M - G -SSet τ is tensored and cotensored over SSet with the tensoring given by the tensoring on E M - G -SSet τ and the cotensoringgiven by applying (–) τ to the usual cotensoring. It is then obvious that ev ω preservestensors, so that (1 . .
13) is a simplicial adjunction. Thus, the positive G -globalmodel structure on E M - G -SSet τ is simplicial.It is clear from the construction that (–) • is right Quillen so that (1 . .
13) is aQuillen adjunction. Moreover, both adjoints preserve and reflect weak equivalencesby definition and the above characterization of the weak equivalences. To show that(1 . .
13) is a Quillen equivalence it is therefore enough that the counit X • ( ω ) → X is a weak equivalence for each tame E M - G -simplicial set X , but we already knowthat it is even an isomorphism.Next, let I and J be sets of generating cofibrations and generating acyclic cofi-brations, respectively, of the positive G -global model structure on G - I -SSet . Weclaim that the positive G -global model structure on E M - G -SSet τ is cofibrantlygenerated (hence combinatorial) with generating cofibrations I ( ω ) := { i ( ω ) : i ∈ I } and generating acyclic cofibrations J ( ω ). Indeed, I ( ω ) and J ( ω ) permit the smallobject argument as E M - G -SSet τ is locally presentable, they consist of cofibra-tions and acyclic cofibrations, respectively, as (1 . .
13) is a Quillen adjunction, andthey detect acyclic fibrations and fibrations, respectively, by adjointness. Taking I to be the usual set of generating cofibrations and using the canonical isomorphism I ( A, –)( ω ) ∼ = E Inj(
A, ω ) shows that (1 . .
11) is a set of generating cofibrations.Next, let us show that (1 . .
12) is a simplicial Quillen adjunction. It is obviousthat the left adjoint preserves tensors, so that this is indeed a simplicial adjunction.Moreover, incl sends the above generating cofibrations to injective cofibrations, andit is homotopical by the above characterization of the weak equivalences, hence leftQuillen. Finally, we observe that in the diagram of homotopical functors G - I -SSet ev ω −−→ E M - G -SSet τ ֒ → E M - G -SSet the left hand functor induces an equivalence on homotopy categories by the aboveand so does the composition by Theorem 1.3.30. Thus, also incl descends to anequivalence on homotopy categories, i.e. (1 . .
12) is a Quillen equivalence. .4. TAMENESS 101
It only remains to show that E M - G -SSet τ is left proper. But indeed, wehave seen that the weak equivalences are precisely the G -global weak equivalences,and since (1 . .
12) is a Quillen adjunction, the cofibrations are in particular injec-tive cofibrations. As pushouts in E M - G -SSet τ can be computed inside all of E M - G -SSet the claim therefore follows from the left properness of the injective G -global model structure on E M - G -SSet . (cid:3) Remark . While we will be almost exclusively interested in the G -globalpositive model structure considered above, the analogous statements for the G -global model structure and the restricted G -global model structure on G - I -SSet hold and can be proven in the same way. We call the resulting model structures the G -global model structure and the restricted G -global model structure , respectively.Their generating cofibrations are again given by evaluating the usual generatingcofibrations for the corresponding model structures on G - I -SSet at ω . Remark . If X is fibrant in the G -global positive model structure, thenall its simplices have ‘small support up to weak equivalence.’ More precisely, let H ⊂ M be universal, and let A ⊂ ω be a non-empty faithful H -set. Then the H -action on X restricts to X [ A ] , and we claim that the inclusion X [ A ] ֒ → X is a G H,G -weak equivalence. Indeed, by definition X • is fibrant in the positive G -globalmodel structure on G - I -SSet , hence in particular positively static.Of course, the analogous statement for the G -global and restricted G -globalmodel structure hold.For later use, we record two properties of the above cofibrations: Lemma . Let f : X → Y be a map in E M - G -SSet . (1) If f is a cofibration in the G -global positive model structure, then f re-stricts to an isomorphism f [ ∅ ] : X [ ∅ ] → Y [ ∅ ] . (2) If f is a G -global cofibration (for example, if f is a restricted or positive G -global cofibration), then it is also a G H,G -cofibration for any subgroup H ⊂ M , i.e. f is injective and G acts freely on Y outside the image of f . Proof.
For the first statement we observe that f • is a G -global positive cofi-bration by definition, so f [ ∅ ] = f • ( ∅ ) is an isomorphism by Lemma 1.3.41.For the second statement, it suffices to prove this for the generating cofibra-tions. As the G H,G -model structure is simplicial, it suffices further to show that E Inj(
A, ω ) × ϕ G is cofibrant in the G H,G -model structure, i.e. that G acts freely onit. This is immediate from Lemma 1.2.33-(2). (cid:3) While the above argument does not apply to the injective G -global model struc-ture, we still have: Theorem . There is a unique model structure on E M - G -SSet τ withcofibrations the injective cofibrations and weak equivalences the G -global weak equiv-alences. This model structure is combinatorial, left proper, simplicial, and filteredcolimits in it are homotopical. Moreover, the simplicial adjunction (1.4.14) ev ω : G - I -SSet inj. G -global ⇄ E M - G -SSet τ inj. G -global : (–) • is a Quillen equivalence. Proof.
Let ˆ I be a set of generating cofibrations for the injective G -globalmodel structure on G - I -SSet , and let I = ˆ I ( ω ) = { i ( ω ) : i ∈ ˆ I } .
02 1. UNSTABLE G -GLOBAL HOMOTOPY THEORY Claim.
The I -cofibrations are precisely the injective cofibrations. Proof.
It is clear that any element of I is an injective cofibration. As thelatter are closed under pushouts, retracts, and transfinite composition, it sufficesto show conversely that any injective cofibration is a retract of an I -cell complex.But indeed, if f is an injective cofibration, then so is f • by direct inspection,so that it can be written as a retract of a relative ˆ I -cell complex. As ev ω is a leftadjoint, we conclude that f • ( ω ) is an ˆ I ( ω ) = I -cofibration. But by full faithfulnessof (–) • , f • ( ω ) is conjugate to f , which completes the proof. △ We have seen in the proof of the previous theorem that pushouts along injec-tive cofibrations in E M - G -SSet preserve G -global weak equivalences, and thatpositive G -global cofibrations are in particular G -global cofibrations. As the injec-tive cofibrations are generated by the set I , Corollary A.2.10 therefore shows thatthe model structure exists, and that is combinatorial, left proper, and that filteredcolimits in it are homotopical.To prove that the model structure is simplicial, it suffices that colimits andtensors can be computed in all of E M - G -SSet and that the latter is simplicial byCorollary 1.1.39.Finally, it is clear that ev ω preserves injective cofibrations and weak equiva-lences, so that (1 . .
14) is a Quillen adjunction, hence a Quillen equivalence by theprevious theorem. (cid:3)
Theorem . There exists a model structure on M - G -SSet τ in which amap f is a weak equivalence, fibration, or cofibration, if and only if f • is a weakequivalence, fibration, or cofibration, respectively, in the positive G -global modelstructure on G - I -SSet . We call this the positive G -global model structure . Itsweak equivalences are precisely the G -global weak equivalences.This model structure is proper, simplicial, combinatorial with generating cofi-brations { (Inj( A, ω ) × ϕ G ) × ∂ ∆ n ֒ → (Inj( A, ω ) × ϕ G ) × ∆ n : H finite group, A = ∅ finite faithful H -set, ϕ : H → G homomorphism } , and filtered colimits in it are homotopical. Finally, the simplicial adjunctions incl : M - G -SSet τ ⇄ M - G -SSet injective G -global : (–) τ ev ω : G - I -SSet ⇄ M - G -SSet τ : (–) • are Quillen equivalences. Proof.
This is proven in the same way as the previous theorem, except thatthe identification of the weak equivalences and the fact that ev ω : G - I -SSet → G - M -SSet is an equivalence on homotopy categories now rely on much harderresults (Theorem 1.4.27 and Corollary 1.4.28, respectively). (cid:3) Corollary . The simplicial adjunction E M × M – : M - G -SSet τ ⇄ E M - G -SSet τ : forget is a Quillen equivalence. Proof.
We have seen in Theorem 1.4.22 that both adjoints are homotopicaland descend to equivalences on homotopy categories. It only remains to show .4. TAMENESS 103 that E M × M – sends the above generating cofibrations to cofibrations, which isimmediate from Corollary 1.4.19. (cid:3) Remark . Again we get analogous statements for the G -global or re-stricted G -global model structure. Moreover, one can construct an injective G -global model structure by an argument similar to the above. We leave the detailsto the interested reader.Finally, let us discuss functoriality for E M - G -SSet τ : Lemma . Let α : H → G be any group homomorphism. Then α ! : E M - H -SSet ⇄ E M - G -SSet : α ∗ restricts to a Quillen adjunction (1.4.15) α ! : E M - H -SSet τ ⇄ E M - G -SSet τ : α ∗ . The right adjoint is fully homotopical. Moreover, if α is injective, then so is theleft adjoint. Proof.
It is clear that α ∗ preserves tameness, and so does α ! as the full sub-category E M -SSet τ ⊂ E M -SSet is closed under colimits.To see that (1 . .
15) is a Quillen adjunction with homotopical right adjoint,it suffices to observe that α ∗ commutes with (–) • on the nose, so that the claimfollows from Lemma 1.3.52.Finally, if α is injective, then α ! sends H -global weak equivalences to G -globalweak equivalences by Corollary 1.2.70. (cid:3) Corollary . In the above situation, α ! preserves H -global weak equiva-lences between objects with free ker( α ) -action. Proof. As α ! can be computed in E M - G -SSet , this is an instance of Corol-lary 1.2.72. (cid:3) The situation for right adjoint is a bit more complicated: α ∗ always has aright adjoint α ∗ by the Special Adjoint Functor Theorem, and α ∗ ⊣ α ∗ is a Quillenadjunction for the injective model structures. However, α ∗ : E M - H -SSet τ → E M - G -SSet τ will usually not agree with α ∗ : E M - H -SSet → E M - G -SSet asthe inclusion E M -SSet τ ֒ → E M -SSet does not preserve all limits. Neverthelesswe have: Lemma . If α : H → G is injective with ( G : im α ) < ∞ , then α ∗ : E M - G -SSet ⇄ E M - H -SSet : α ∗ restricts to a Quillen adjunction α ∗ : E M - G -SSet τ ⇄ E M - H -SSet τ : α ∗ in which both functors are fully homotopical. Proof.
We already know that α ∗ preserves tameness and is fully homotopical.To see that also α ∗ preserves tameness, we observe that as an E M -simplicial set, α ∗ X is just a ( G : im α )-fold product of copies of X , and that E M -SSet τ ⊂ E M -SSet is closed under finite limits.It only remains to show that α ∗ preserves fibrations as well as weak equiv-alences. But indeed, as α ∗ commutes with ev ω , α ∗ commutes with (–) ω up to(canonical) isomorphism, so these follow from Lemma 1.3.55. (cid:3)
04 1. UNSTABLE G -GLOBAL HOMOTOPY THEORY By abstract nonsense, (finite) products of fibrations in E M - G -SSet τ are fi-brations, and we have seen that also finite products of G -global weak equivalencesare weak equivalences. If now S is any finite set and X is any E M - G -simplicialset, then X × S := Q s ∈ S X carries a natural Σ S -action by permuting the factors,and this way (–) × S lifts to E M - G -SSet → E M -( G × Σ S )-SSet . We will laterneed the following strengthening of the above observation: Corollary . The above lift of (–) × S sends G -global weak equivalencesor fibrations to ( G × Σ S ) -global weak equivalences or fibrations, respectively. Proof.
The claim is trivial if S is empty. Otherwise, we pick s ∈ S , and writeΣ S ⊂ Σ S for the subgroup of permutations fixing s . We write p : G × Σ S → G forthe projection to the second factor, and i : G × Σ S → G × Σ S for the inclusion. Claim.
The functor (–) × S is isomorphic to i ∗ ◦ p ∗ . Proof.
Fix for each s ∈ S a permutation σ s ∈ Σ S with σ s ( s ) = s . Then weconsider for X ∈ E M - G -SSet the natural map maps G × Σ S ( G × Σ S , X ) → X × S given on the s -th factor by evaluating at (1 , σ − s ). We omit the easy verificationthat this is ( G × Σ S )-equivariant and an isomorphism. △ Thus, the statement follows from Lemmas 1.4.44 and 1.4.46. (cid:3) G -global homotopy theory vs. global homotopy theory In this section we prove as promised that our theory generalizes Schwede’s unstable global homotopy theory with respect to finite groups. More precisely, wewill give a chain of Quillen adjunctions between I -SSet and Schwede’s orthogonalspaces that on associated quasi-categories exhibits the former as a (right Bousfield)localization with respect to an explicit class of ‘ F in -global weak equivalences.’ Orthogonal spaces are basedon a certain topological analogue L of the categories I and I considered above.Explicitly, the objects of L are the finite dimensional real inner product spaces V ,and as a set L ( V, W ) is given by the isometric embeddings V → W . For V = W , L ( V, W ) carries the topology of the orthogonal group O ( V ), and in general L ( V, W )is topologized as a Stiefel manifold; since we can completely blackbox the topology,we omit the details and refer the curious reader to [
Sch18 , Section 1.1] instead.
Definition . An orthogonal space is a topologically enriched functor L → Top . We write L -Top for the topologically enriched functor category Fun ( L, Top ).Schwede [
Sch18 , Definition 1.1.1] denotes the above category by ‘ spc ’ and heconstructs a global model structure on it that we will recall now.1.5.1.1.
The global level model structure.
As for I - and I -simplicial sets, thedesired global model structure on L -Top will be constructed as a Bousfield local-ization of a suitable level model structure. Definition . A map f : X → Y of orthogonal spaces is called a global levelweak equivalence or global level fibration if f ( V ) is a weak equivalence or fibration,respectively, for any V ∈ L in the equivariant model structue on O ( V )-Top withrespect to all closed subgroups, i.e. f ( V ) H is a weak homotopy equivalence orfibration, respectively, for any closed subgroup H ⊂ O ( V ). .5. G -GLOBAL HOMOTOPY THEORY VS. GLOBAL HOMOTOPY THEORY 105 Schwede [
Sch18 , Definition 1.1.8] calls the above ‘strong level equivalences’and ‘strong level fibrations,’ respectively.If G is a compact Lie group, then an orthogonal G -representation is an object V ∈ L together with a continuous homomorphism ρ : G → O ( V ); equivalently, wecan view this as a pair of a finite dimensional real inner product space V togetherwith a continuous G -action by linear isometries. Restricting along ρ , X ( V ) is thennaturally a G -space for every orthogonal space X , and f ( V ) : X ( V ) → Y ( V ) is G -equivariant for any map f : X → Y of orthogonal spaces. The above conditioncan then be rephrased as saying that f ( V ) should be a weak equivalence or fi-bration, respectively, in G -Top for any compact Lie group G and any orthogonal G -representation V , see [ Sch18 , Lemmas 1.2.7 and 1.2.8].
Proposition . There is a unique model structure on L -Top with weakequivalences the global level weak equivalences and fibrations the global level fibra-tions. It is topological and cofibrantly generated with generating cofibrations { L ( V, –) /H × ∂D n ֒ → L ( V, –) /H × D n : V ∈ L, H ⊂ O ( V ) closed , n ≥ } . Proof.
This is [
Sch18 , Proposition 1.2.10] and the discussion after it. (cid:3)
Global weak equivalences.
The global weak equivalences of orthogonalspaces are slightly intricate to define due to some pointset topological issues. In-tuitively speaking, however, they should again be created by ‘evaluating at R ∞ ,’analogously to the approach for I -simplicial sets: Construction . We write L for the monoid of linear isometric embed-dings R ∞ → R ∞ under composition; here the scalar product on R ∞ = R ( ω ) (thevector space of functions ω → R vanishing almost everywhere) is so that the canon-ical basis consisting of the characteristic functions is orthonormal. The topologyon L is given as a subspace of the mapping space.Let X be an orthogonal space. Then [ Sch20 , Construction 2.3] describes how X yields a space X ( R ∞ ) with a continuous L -action. Explicitly, we define X ( R ∞ ) := colim V ⊂ R ∞ finite dimensional X ( V ) , where the structure maps of the colimit system are induced via X from the inclu-sions. A continuous L -action is given as follows: if x is contained in the image of X ( V ) → X ( R ∞ ), then u.x is the image of x under the composition X ( V ) X ( u | V : V → u ( V )) −−−−−−−−−−−→ X ( u ( V )) → X ( R ∞ ) , where the right hand map is the structure map of the colimit.By functoriality of colimits, the above extends to a functor ev R ∞ : L -Top → L -Top .Unlike for simplicial sets, filtered colimits of topological spaces do not preserveweak equivalences in general, which already suggests that ev R ∞ is not homotopicallymeaningful. Schwede [ Sch18 , Definition 1.1.2] avoids this problem by defining theweak equivalences in terms of a ‘homotopy extension lifting property’ instead. Wewill take a different approach following [
Sch20 ] here:Non-equivariantly, we could solve this issue by replacing the above colimitby a homotopy colimit, but this does not retain equivariant information. Namely,whether f : X → Y is a global weak equivalence should of course not only depend on f ( R ∞ ) as a map of ordinary topological spaces, but it should take into account the
06 1. UNSTABLE G -GLOBAL HOMOTOPY THEORY L -action and in particular suitable actions of all compact Lie groups. Concretely,see [ Sch20 , Definitions 1.3, 1.4, and 1.6]:
Definition . A compact subgroup H ⊂ L is called universal if it admitsthe structure of a Lie group and the tautological H -action on R ∞ makes the latterinto a complete H -universe, i.e. any finite dimensional orthogonal H -representationembeds H -equivariantly and isometrically into R ∞ .A map f : X → Y of L -spaces is a global weak equivalence if f H : X H → Y H isa weak homotopy equivalence for every universal H ⊂ L .One way to calculate homotopy colimits is by taking a cofibrant replacementin the projective model structure. In our situation we will instead replace in theglobal level model structure: Definition . An orthogonal space X is called closed if for any map ϕ : V → W in L the induced map X ( ϕ ) : X ( V ) → X ( W ) is a closed embedding. Example . Any orthogonal space that is cofibrant in the global levelmodel structure is closed, see [
Sch18 , Proposition 1.2.11-(iii)].
Definition . Let f : X → Y be a map of orthogonal spaces and letˆ X X ˆ Y Y ˆ f ∼ f ∼ be a commutative diagram such that the horizontal maps are global level weakequivalences and ˆ X, ˆ Y are closed. Then f is called a global weak equivalence ifˆ f ( R ∞ ) is a global weak equivalence of L -spaces.Note that we can indeed always find such a square by simply taking functo-rial cofibrant replacements in the global level model structure. Moreover, [ Sch20 ,Proposition 3.5] together with [
Sch18 , Proposition 1.1.9-(i)] shows that the aboveis independent of the choice of replacement and equivalent to Schwede’s originaldefinition.
Theorem . There is a unique model structure on L -Top with the samecofibrations as the global level model structure and with the global weak equivalencesas weak equivalences. This model structure is topological, proper, and cofibrantlygenerated with generating cofibrations { L ( V, –) /H × ∂D n ֒ → L ( V, –) /H × D n : V ∈ L, H ⊂ O ( V ) closed , n ≥ } . Moreover, an orthogonal space X is fibrant if and only if it is static in the sensethat X ( ϕ ) : X ( V ) → X ( W ) is a G -equivariant weak equivalence for any compact Liegroup G and any G -equivariant linear isometric embedding ϕ : V → W of faithful finite dimensional orthogonal G -representations. Proof.
This is [
Sch18 , Theorem 1.2.21]. (cid:3)
Again, there is also a positive global model structure where one restricts thegenerating cofibrations by demanding in addition that V = 0, see [ Sch18 , Propo-sition 1.2.23].By design, our models of global homotopy theory only see equivariant informa-tion with respect to finite groups, so we should not hope for them to be equivalent .5. G -GLOBAL HOMOTOPY THEORY VS. GLOBAL HOMOTOPY THEORY 107 to the above model category. Instead, we consider the following coarser notion ofweak equivalence: Definition . A map of L -spaces is called a F in -global weak equivalence if it restricts to a G -equivariant weak equivalence for each finite universal G ⊂ L .A map f of orthogonal spaces is called a F in -global weak equivalence if ˆ f ( R ∞ )is a F in -global weak equivalence in L -Top for some (hence any) replacement of f by a map ˆ f between closed orthogonal spaces. Remark . As remarked without proof in [
Sch20 , Remark 3.11], there isa version for the global model structure on L -Top which only sees representationsof groups belonging to a given global family F , i.e. a collection of compact Liegroups closed under isomorphisms and subquotients. For F = F in the family offinite groups this precisely recovers the above F in -global weak equivalences. I -spaces. The intermediate step in our comparison will be a globalmodel structure on I -Top . For this the following terminology will be useful, see[ Sch18 , discussion after Definition A.28]:
Definition . Let C be a category enriched and tensored over Top . Thena map f : A → B in C is an h-cofibration if the natural map (cid:0) A × [0 , (cid:1) ∐ A B → B × [0 ,
1] from the mapping cyclinder of f admits a retraction. Example . For C = Top with the usual enrichment, the h-cofibrationsare the classical (Hurewicz) cofibrations.
Lemma . Let G be a finite group and let (1.5.1) A BC D i be a pushout in G -Top such that i is an h-cofibration. Then Sing : G -Top → G -SSet takes (1 . . to a homotopy pushout. Proof.
Let H ⊂ G be any subgroup. The functor (–) H preserves pushoutsalong closed embeddings by [ Sch18 , Proposition B.1-(i)] and it clearly preservestensors; it easily follows that it preserves h-cofibrations, also cf. [
Sch19b , Corol-lary A.30-(ii)]. We conclude that i H is a Hurewicz cofibration and that the squareon the left in A H B H C H D Hi H (Sing A ) H (Sing B ) H (Sing C ) H (Sing D ) H (Sing i ) H is a pushout in Top . It is well-known that Sing sends pushouts along Hurewiczcofibrations to homotopy pushouts, and as it moreover commutes with fixed points,we conclude that the square on the right is a homotopy pushout in
SSet . The claimnow follows from Theorem 1.1.31. (cid:3)
Corollary . Let i : A → B be any h-cofibration of I -spaces. Then Sing : I -Top → I -SSet sends pushouts along i to homotopy pushouts in the globalmodel structure.
08 1. UNSTABLE G -GLOBAL HOMOTOPY THEORY Proof.
The evaluation functors I -Top → Top are cocontinuous and pre-serve tensors, so h-cofibrations of I -spaces are in particular levelwise h-cofibrationsby [ Sch18 , Corollary A.30-(ii)]. Applying the previous lemma levelwise thereforeshows that the canonical comparison map Sing( B ) ∐ Sing( A ) Sing( C ) → Sing( B ∐ A C )(for any map A → C ) is a global level weak equivalence, hence in particular a globalweak equivalence. On the other hand, Sing( i ) is an injective cofibration, so that theordinary pushout Sing( B ) ∐ Sing( A ) Sing( C ) is a homotopy pushout by the existenceof the injective model structure on I -SSet . The claim follows immediately. (cid:3) Proposition . There is a unique model structure on I -Top in which amap f is a weak equivalence or fibration if and only if Sing f is a weak equivalenceor fibration, respectively, in the global model structure on I -SSet . We call this the global model structure and its weak equivalences the global weak equivalences. An I -space X is fibrant in this model structure if and only if it is static in the sensethat X ( i ) H : X ( A ) H → X ( B ) H is a weak homotopy equivalence for any finite group H and any H -equivariant injection i : A → B of finite faithful H -sets.The global model structure is topological, left proper, and cofibrantly generatedwith generating cofibrations I ( A, –) /H × ∂D n ֒ → I ( A, –) /H × D n , where A and H are as above and n ≥ . Moreover, pushouts along h-cofibrationsare homotopy pushouts.Finally, the adjunction (1.5.2) | – | : I -SSet global ⇄ I -Top global : Sing is a Quillen equivalence in which both adjoints are homotopical. Proof.
The adjunction | – | : SSet ⇄ Top : Sing is a Quillen equivalence inwhich both adjoints are homotopical. In particular, the unit η : X → Sing | X | is aweak homotopy equivalence for any space X . As both adjoints commute with finitelimits, it follows further that for any group K acting on X , the unit η X inducesweak homotopy equivalences on K ′ -fixed points for all finite K ′ ⊂ K , so that theunit of (1 . .
2) is a global level weak equivalence for all X ∈ I -SSet .Let us now construct the model structure, for which we will verify the con-ditions of Crans’ Transfer Criterion (Proposition 1.1.5): we let I be the usual setof generating cofibrations, and we pick any set J of generating acyclic cofibrationsof I -SSet global . As each cofibration is in particular a levelwise injection, we con-clude that | I | and | J | consist of closed embeddings. As Sing preserves transfinitecompositions along closed embeddings, we conclude from the local presentability of I -SSet that | I | and | J | permit the small object argument.It remains to show that any relative | J | -cell complex is sent to a global weakequivalence under Sing. Again using that Sing preserves transfinite compositionsalong closed embeddings, it suffices to show that pushouts of maps in | J | are globalweak equivalences.It is clear that the maps in | I | are h-cofibrations, hence so is the geometricrealization of any cofibration by [ Sch18 , Corollary A.30-(i)], and in particular anymap in | J | . As the unit is a levelwise global weak equivalence, all maps in | J | aresent to global weak equivalences under Sing, and hence so is any pushout of a mapin | J | by Corollary 1.5.15. This completes the proof of the existence of the modelstructure. Moreover, as the unit is obviously a levelwise global weak equivalence .5. G -GLOBAL HOMOTOPY THEORY VS. GLOBAL HOMOTOPY THEORY 109 and as the right adjoint creates weak equivalences by definition, we immediatelyconclude that (1 . .
2) is a Quillen equivalence and that also | – | is homotopical.As seen above, pushouts along h-cofibrations preserve weak equivalences; sincethe cofibrations are h-cofibrations by another application of [ Sch18 , Corollary A.30-(i)], we in particular see that I -Top is left proper. Again using that h-cofibrationsare closed under pushouts by loc.cit. , one easily concludes that pushouts alongh-cofibrations are homotopy pushouts.As a functor category, I -Top is enriched, tensored, and cotensored over Top in the obvious way. Restricting along the adjunction | – | : SSet ⇄ Top : Singtherefore makes it into a category enriched, tensored, and cotensored over
SSet .With respect to this, (1 . .
2) is naturally a simplicial adjunction, so I -Top is asimplicial model category by Lemma 1.1.4-(2). To see that it is topological, we thensimply observe that it suffices to verify the pushout product axiom for generatingcofibrations and generating (acyclic) cofibrations, and that the ones for Top agreeup to conjugation by isomorphisms with the images of the ones of
SSet undergeometric realization. (cid:3)
Remark . In analogy with Definition 1.5.6 let us call an I -space X closed if all structure maps X ( A ) → X ( B ) are closed embeddings. As Sing preservessequential colimits along closed embeddings, it easily follows that Sing commuteswith ev ω on the subcategory of closed I -spaces. In particular, if f : X → Y is amap of closed I -spaces that is a weak equivalence at ∞ in the sense that f ( ω ) H is aweak homotopy equivalence for any universal H ⊂ M , then f is already a G -globalweak equivalence. In order to relate I -spaces to orthogonalspaces, we introduce: Construction . We define a functor R • : I → L as follows: a finite set A is sent to the real vector space R A of maps A → R , where the inner product on R A is the unique one such that the characteristic functions of elements of A forman orthonormal basis.If f : A → B is an injection of sets, then R f : R A → R B is defined to be theunique R -linear map sending the characteristic function of a ∈ A to the character-istic function of f ( a ) ∈ B . It is clear that R f is isometric, and that this makes R • into a well-defined functor.Restricting along the above yields a forgetful functor L -Top → I -Top . Bytopologically enriched left Kan extension, this admits a topological left adjoint L × I – satisfying L × I I ( A, –) = L ( A, –) for any finite set A ; the unit is then givenon such representables by R • : I ( A, –) → L ( R A , R • ) = forget L ( R A , –). Remark . Let H be a finite group and let U be a countable set containinginfinitely many free H -orbits (for example if U is a complete H -set universe). Thenthe R -linearization R ( U ) contains infinitely many copies of the regular representation R H , so it is a complete H -universe in the usual sense. In particular, if H ⊂ M is universal, then the induced H -action on R ∞ = R ( ω ) makes the latter into acomplete H -universe. Proposition . The map R • : I ( A, –) /H × X → L ( R A , R • ) /H × X is aglobal weak equivalence in I -Top for any finite group H , any finite faithful H -set A , and any CW-complex X .
10 1. UNSTABLE G -GLOBAL HOMOTOPY THEORY Proof.
Taking products with X obviously preserves global level weak equiv-alences, and it preserves global acyclic cofibrations as I -Top is topological. Thus, X × – is fully homotopical, and we are reduced to X = ∗ .For this we consider the commutative diagram(1.5.3) |I ( A, –) | /HI ( A, –) /H (cid:0) |I ( A, –) | × L ( R A , R • ) (cid:1) /HL ( R A , R • ) /H prpr where the maps from left to right are induced by the inclusion I ( A, –) ֒ → |I ( A, –) | and by R • : I ( A, –) → L ( R A , R • ). We begin by showing that the vertical arrows onthe right are global weak equivalences.It is clear that the I -space |I ( A, –) | is closed, and so is L ( R A , R • ) by Exam-ple 1.5.7. From this one concludes by [ Sch18 , Proposition B.13-(iii)] that all the I -spaces on the right of (1 . .
3) are closed, so that it suffices that the vertical mapsare weak equivalences at ∞ .Let K ⊂ M be a universal subgroup. Obviously, |I ( A, ω ) | is an ( H × K )-CWcomplex, and so is L ( R A , R ∞ ) by [ Sch18 , Proposition 1.1.19-(ii)]. We claim thatthey are both classifying spaces for G K,H in the sense that their T -fixed pointsfor T ⊂ K × H are contractible when T ∈ G K,H , and empty otherwise. Indeed,for L ( R A , R ∞ ) this is an instance of [ Sch18 , Proposition 1.1.26-(i)]. On the otherhand, if K ′ ⊂ K , ϕ : K ′ → H , then to show that |I ( A, ω ) | ϕ ∼ = | E (Inj( A, ω ) ϕ ) | iscontractible it suffices that there exists a K ′ -equivariant injection ϕ ∗ A → ω , whichis immediate from universality. Finally, the ( K × H )-equivariance of R • : I ( A, ω ) → L ( R A , R ∞ ) shows that I ( A, ω ) T has no vertices for T ⊂ K × H not in G K,H , so ithas to be empty.Thus, the projections |I ( A, ω ) | ← |I ( A, ω ) | × L ( R A , R ∞ ) → L ( R A , R ∞ ) areweak equivalences of cofibrant objects in the G K,H -equivariant model structure on ( K × H )-Top . Applying Ken Brown’s Lemma to the Quillen adjunction p ! : ( K × H )-Top G K,H -equivariant ⇄ K -Top A ℓℓ : p ∗ where p : K × H → K denotes the projection, therefore shows that after evaluationat ∞ the vertical maps in (1 . .
3) become G K,G -weak equivalences. The claim followsby letting K vary.To finish the proof we observe now that I ( A, –) /H → |I ( A, –) | /H agrees up toconjugation by isomorphisms with the image of(1.5.4) I ( A, –) /H ֒ → I ( A, –) /H under geometric realization. The proposition follows as (1 . .
4) is a global weakequivalence by Theorem 1.4.29 and since geometric realization is fully homotopicalby Proposition 1.5.16. (cid:3)
We define a monoid homomorphism i : M → L by sending f : ω → ω to R f : R ∞ → R ∞ , i.e. the unique linear isometry sending the i -th standard basisvector to the f ( i )-th one. Lemma . Let X be an L -space. Then Sing( i ∗ X ) ∈ M -SSet is semistable. .5. G -GLOBAL HOMOTOPY THEORY VS. GLOBAL HOMOTOPY THEORY 111 Proof.
Let H ⊂ M be universal, and let u ∈ M centralize H . It suffices toshow that i ( u ) . – : X → X is an i ( H )-equivariant weak equivalence.Indeed, Remark 1.5.19 implies that i ( H ) is a universal subgroup of L , so L i ( H ) is contractible by [ Sch18 , Proposition 1.1.26-(i)] together with [
Sch20 , Proposi-tion A.10]. In particular, there exists a path γ : [0 , → L i ( H ) connecting i ( u ) tothe identity. The composition[0 , × X γ × X −−−→ L × X action −−−−→ X is then an i ( H )-equivariant homotopy from i ( u ) . – to the identity, so i ( u ) . – is inparticular an i ( H )-equivariant (weak) homotopy equivalence. (cid:3) Proposition . Let f : X → Y be a map of orthogonal spaces. Then thefollowing are equivalent: (1) f is a F in -global weak equivalence (Definition 1.5.10). (2) forget f is a global weak equivalence of I -spaces.Moreover, if X and Y are closed, then also the following statements are equivalentto the above: (3) Sing (cid:0) i ∗ f ( R ∞ ) (cid:1) is a universal weak equivalence of M -simplicial sets. (4) Sing (cid:0) i ∗ f ( R ∞ ) (cid:1) is a global weak equivalence of M -simplicial sets. Proof.
Let us first assume that X and Y are closed; we will show that all ofthe above statements are equivalent.For the equivalence (1) ⇔ (3) we observe once more that i : M → L sendsuniversal subgroups to universal subgroups. As any two abstractly isomorphicuniversal subgroups of L are conjugate [ Sch20 , Proposition 1.5], the claim nowfollows from the definitions.The equivalence (3) ⇔ (4) is immediate from the previous lemma. Finally, for(2) ⇔ (4) it suffices by Remark 1.5.17 together with Theorem 1.4.27 to show thatthe diagram L -Top L -Top I -Top M -Top ev R ∞ forget i ∗ ev ω commutes up to natural isomorphism. This follows easily from the definitions oncewe observe that the map of posets { A ⊂ ω finite } → { V ⊂ R ∞ finite dimensional } sending A to the image of the canonical map R A → R ∞ is cofinal: if V ⊂ R ∞ ,then there is only a finite set S ( v ) of i ∈ ω such that pr i ( v ) = 0 for the projectionpr i : R ∞ → R to the i -th summand. Thus, if V ⊂ R ∞ is any finite dimensionalsubspace, and v , . . . , v n is a basis, then S := S ( v ) ∪ · · · ∪ S ( v n ) is finite, andobviously V is contained in the image of R S → R ∞ .Now assume X and Y are not necessarily closed. If j : A → B is a global levelweak equivalence of orthogonal spaces, then j is in particular a F in -global weakequivalence; moreover Sing(forget( j )) is obviously a global level weak equivalenceof I -simplicial sets, hence in particular a global weak equivalence. Thus, (1) ⇔ (2)follows from the above special case together with 2-out-of-3. (cid:3) Theorem . The topologically enriched adjunction (1.5.5) L × I – : I -Top ⇄ L -Top : forget
12 1. UNSTABLE G -GLOBAL HOMOTOPY THEORY is a Quillen adjunction. The induced adjunction of associated quasi-categories is aright Bousfield localization with respect to the F in -global weak equivalences. Proof.
Let us first show that (1 . .
5) is a Quillen adjunction. As it is a topolog-ically enriched adjunction of topological model categories, it in particular becomesa simplicial adjunction of simplicial model categories when we restrict along theusual adjunction
SSet ⇄ Top . It therefore suffices (Proposition A.2.8) to observethat L × I – obviously preserves generating cofibrations, and that the forgetful func-tor preserves fibrant objects by the characterizations given in Theorem 1.5.9 andProposition 1.5.16, respectively.By the previous proposition, the forgetful functor is homotopical and it preciselyinverts the F in -global weak equivalences. It therefore only remains to show that theunit X → forget( L × I X ) is a global weak equivalence for any cofibrant X ∈ I -Top .This will again be a cell induction argument (though we cannot literally applyCorollary 1.1.8): if X is the source or target of one of the standard generatingcofibrations of I -Top , then the claim is an instance of Proposition 1.5.20. On theother hand, any pushout I ( A, –) /H × ∂D n − I ( A, –) /H × D n X Y along a generating cofibration is a homotopy pushout by left properness, and itsimage under forget ◦ ( L × I –) is a pushout along an h-cofibration by [ Sch18 , Corol-lary A.30-(ii)], hence again a homotopy pushout by Proposition 1.5.16 above. Weconclude that η Y is a global weak equivalence if η X is.Using that transfinite compositions of closed embeddings in I -Top are homo-topical, we therefore see that η Z is a global weak equivalence for any cell complex Z in the generating cofibrations. The claim follows as any cofibrant object of I -Top is a retract of such a cell complex. (cid:3) Corollary . The functor
Sing ◦ forget : L -Top → I -SSet preservesglobal fibrations and global weak equivalences, and it induces a quasi-localization L -Top ∞ → I -SSet ∞ . (cid:3) Remark . It is not hard to show that (1 . .
5) is a Quillen adjunction withrespect to the F in -global model structure on L -Top mentioned in Remark 1.5.11,hence a Quillen equivalence by the above theorem.On the other hand, one can easily adapt the above proof to transfer the globalmodel structure from I -Top to L -Top . This way one obtains a model structurewith the F in -global weak equivalences as global weak equivalences but slightlyfewer cofibrations than the model structure sketched by Schwede.HAPTER 2 Coherent commutativity
In this chapter we introduce several models of ‘ G -globally coherently com-mutative monoids,’ either based on the notion of ultra-commutativity studied bySchwede [ Sch18 ] in the global context, or on Γ -spaces , originally introduced bySegal [
Seg74 ] and then later generalized equivariantly by Shimakawa [
Shi89 ]. Asour main result (Theorem 2.3.1) in this chapter we prove that all these approachesare equivalent.
In this section we will study various box products for our different models of G -global homotopy theory, and in particular we will revisit the box products on M -SSet and I -SSet of [ SS20 ] from a G -global perspective. We will show that allthese box products are in fact fully homotopical, which among other things yields G -global versions of [ SS20 , Theorem 1.2 and Theorem 4.8].Our main goal then is to lift the equivalences of unstable G -global homotopytheory to equivalences between the corresponding homotopy theories of commu-tative monoids. In particular this will give a G -global refinement of the equiv-alence between the non-equivariant homotopy theories of CMon( M -SSet τ ) andCMon( I -SSet ) due to Schwede and Sagave, see [ SS20 , proof of Theorem 5.13].
We briefly recall the box products of M - and I -simplicial sets, which will serve as blueprints for the definition of the boxproducts of E M - and I -simplicial sets. Construction . The coproduct of (finite) sets enhances I to a symmetricmonoidal category. More precisely, we have a functor I × I → I given on objectsby (
A, B ) A ∐ B , and on morphisms by sending a pair f : A → A ′ , g : B → B ′ to f ∐ g : A ∐ B → A ′ ∐ B ′ . The usual unitality, associativity, and symmetryisomorphisms of the cocartesian symmetric monoidal structure on Set then inducenatural isomorphisms for the corresponding functors on I , and these satisfy theusual coherence condition.This symmetric monoidal structure then induces a Day convolution product on I -SSet as follows: by the universal property of enriched presheaves, there is a sim-plicially enriched functor – ⊠ – : I -SSet × I -SSet → I -SSet that preserves tensorsand small colimits in each variable separately and such that I ( A, –) ⊠ I ( B, –) = I ( A ∐ B, –) with the evident functoriality in A and B . The functor ⊠ is uniqueup to a unique simplicial isomorphism that is the identity on pairs of representa-bles. We fix such a choice and call it ‘the’ box product on I -SSet . This isthe symmetric monoidal product of a preferred simplicial symmetric monoidalstructure on I -SSet , where all the structure isomorphisms are induced from the corresponding structure isomorphisms of ∐ . More precisely, the symmetry iso-morphism τ is the unique simplicially enriched natural isomorphism such that τ : I ( A, –) ⊠ I ( B, –) → I ( B, –) ⊠ I ( A, –) agrees for all finite A, B with restrictionalong the inverse of the flip A ∐ B → B ∐ A , and similarly for associativity andunitality.Sagave and Schlichtkrull [ SS12 , Theorem 1.2] showed that strictly commutativemonoids for the box product on I -sets model all coherently commutative monoidsin non-equivariant spaces. Moreover, Schwede [ Sch18 , Chapter 2] used a variant ofthis for orthogonal spaces as the basis for his approach to coherent commutativityin the global context; we will revisit his construction later in Subsection 2.1.6.Next, we come to the box product of M -sets and M -simplicial sets, which wasintroduced and studied in [ SS20 ]. Definition . Let
X, Y be tame M -sets. Their box product is the M -subset X ⊠ Y ⊂ X × Y consisting of precisely those ( x, y ) ∈ X × Y such thatsupp( x ) ∩ supp( y ) = ∅ .By [ SS20 , Proposition 2.13] this is indeed a M -subset of X × Y ; moreover, itbecomes a subfunctor of the cartesian product, and the usual unitality, associativity,and symmetry isomorphisms restrict to make the box product the tensor productof a preferred symmetric monoidal structure on M -Set [ SS20 , Proposition 2.14].Applying the box product levelwise and pulling through the G -actions we thereforealso get a box product on M - G -SSet . Example . The map Inj( A ∐ B, ω ) → Inj(
A, ω ) → Inj(
B, ω ) inducedby the restrictions factors through an isomorphism Inj( A ∐ B, ω ) ∼ = Inj( A, ω ) ⊠ Inj(
B, ω ), see [
SS20 , Example 2.15].
Remark . In [
SS20 , Proposition 4.7], Sagave and Schwede used theabove example to construct a preferred strong symmetric monoidal structure onev ω : I -SSet → M -SSet ; it follows formally that the right adjoint (–) • acquires alax symmetric monoidal structure (which happens to be strong in this case) givenby the composites X • ⊠ Y • η −→ ( X • ⊠ Y • )( ω ) • ψ − • −−−→ ( X • ( ω ) ⊠ Y • ( ω )) • ǫ ⊠ ǫ −−→ ( X ⊠ Y ) • i.e. the canonical mates of the inverse structure isomorphism of ev ω , and the uniquemap ∗ → ∗ • . In particular, we get an induced adjunctionev ω : CMon( I -SSet ) ⇄ CMon( M -SSet ) : (–) • and Sagave and Schwede show in [ SS20 , proof of Theorem 5.13] that this is anequivalence of homotopy theories with respect to suitable weak equivalences. E M -simplicial sets. In order to define thebox product of E M - G -simplicial sets as introduced in [ Len20a ] we need a finernotion of support:
Definition . Let X be an E M -simplicial set, and let 0 ≤ k ≤ n . Thenwe say that x ∈ X n is k -supported on a finite set A ⊂ ω if it is supported on A asan element of the M -set i ∗ k X n where i k : M → M n +1 denotes the inclusion of the( k + 1)-th factor. We call x k -finitely supported if it is k -supported on some finiteset A , in which case we define its k -support supp k ( x ) as its support as an elementof i ∗ k X n . .1. ULTRA-COMMUTATIVITY 115 The following lemma in particular shows that our notions of support and tame-ness (Definition 1.4.14) agree with the ones considered in [
Len20a ]: Lemma . Let X be an E M -simplicial set, n ≥ , and x ∈ X n . Then X is supported on the finite set A ⊂ ω if and only if it is k -supported on A forall ≤ k ≤ n . In particular, x is finitely supported if and only if it is k -finitelysupported for all ≤ k ≤ n , in which case supp( x ) = S nk =0 supp k ( x ) . Proof.
It suffices to prove the first statement. For this we observe that if x is k -supported on A for all 0 ≤ k ≤ n , then u.x = i ( u ) .i ( u ) . . . . .i n ( u ) .x = x for all u ∈ M A by an immediate inductive argument, i.e. x is supported on A byTheorem 1.4.15. Conversely, if x is supported on A and u ∈ M A , then i k ( u ) ∈M n +1 A , so i k ( u ) .x = x by definition. (cid:3) Definition . Let
X, Y be tame E M -simplicial sets. Their box product X ⊠ Y is the subcomplex of X × Y whose n -simplices are precisely those pairs( x, y ) ∈ X n × Y n such that supp k ( x ) ∩ supp k ( y ) = ∅ for all 0 ≤ k ≤ n .This is indeed a subcomplex and closed under the E M -action by [ Len20a ,Proposition 2.17], and we have in addition shown at loc.cit. that the box productdefines a subfunctor of the cartesian product on E M -SSet τ . Pulling through theactions, we can extend this to E M - G -SSet τ , and by [ Len20a , Proposition 2.18]the unitality, associativity, and symmetry isomorphisms of the cartesian productrestrict to yield a preferred symmetric monoidal structure on E M - G -SSet τ . Example . Let
A, B be finite sets. One easily checks that the map E Inj( A ∐ B, ω ) → E Inj(
A, ω ) × E Inj(
B, ω ) induced by the inclusions
A ֒ → A ∐ B ← ֓B restricts to E Inj( A ∐ B, ω ) ∼ = E Inj(
A, ω ) ⊠ E Inj(
B, ω ); we also give a completeproof as [
Len20a , Lemma 5.2-(4)].
Definition . We write G -ParSumSSet := CMon( E M - G -SSet τ ) andcall its objects G -parsummable simplicial sets . Remark . Let X , . . . , X n be tame E M -simplicial sets. One easily showsby induction that the image of the canonical injection X ⊠ · · · ⊠ X n → Q ni =1 X i is independent of the chosen bracketing on the left and consists in degree m ofprecisely those ( x , . . . , x n ) such that supp k ( x i ) ∩ supp k ( x j ) = ∅ for all 0 ≤ k ≤ m and 1 ≤ i < j ≤ n . We will from now on use the notation X ⊠ · · · ⊠ X n for this‘unbiased’ iterated box product.2.1.2.1. Homotopical properties.
We will now study the behaviour of the G -global weak equivalences under the above box product. While the proofs are similarto the categorical situation considered for G = 1 in [ Sch19b ], we neverthelessinclude them for completeness.
Theorem . Let
X, Y ∈ E M - G -SSet τ . Then the inclusion X ⊠ Y ֒ → X × Y is a G -global weak equivalence. In particular, ⊠ preserves G -global weakequivalences in each variable. For the proof we will use:
Lemma . Let X be a tame E M -simplicial set, let ≤ k ≤ n , let x ∈ X n ,and let u , . . . , u n ∈ M . Then supp k (( u , . . . , u n ) .x ) = u k (supp k ( x )) .
16 2. COHERENT COMMUTATIVITY
Proof.
We have ( u , . . . , u n ) .x = i ( u ) . . . . .i n ( u n ) .x . For each ℓ = k , the map i ℓ ( u ℓ ) . – : i ∗ k ( X n ) → i ∗ k ( X n ) is M -equivariant, and it is moreover injective by tame-ness of i ∗ ℓ X . It follows that they preserve k -supports, so that supp k (( u , . . . , u n ) .x ) =supp k ( i k ( u k ) .x ). The claim now follows from Lemma 1.4.4. (cid:3) Proof of Theorem 2.1.11.
It suffices to prove the first statement. We let H ⊂ M be any universal subgroup and we will show that the inclusion i is an( H × G )-equivariant homotopy equivalence.To this end we choose an H -equivariant isomorphism ω ∼ = ω ∐ ω , yielding α, β ∈M centralizing H and such that im( α ) ∩ im( β ) = ∅ . We define r : X × Y → X ⊠ Y via ( x, y ) ( α.x, β.y ) for all ( x, y ) ∈ ( X × Y ) n . This is well-defined by the previouslemma; it is moreover obviously G -equivariant and it is H -equivariant as α and β centralize H .The composition ri is by definition given by ( α. –) × ( β. –) : X × Y → X × Y .Acting with ( α, ∈ E M on X yields a homotopy ∆ × X → X from the identityto α. –; analogously, ( β, ∈ E M yields a homotopy ∆ × Y → Y from the identityto β. –. Altogether we get a homotopy Φ from the identity to ri as desired.To show that also ir is homotopic to the identity, we only have to show that Φrestricts to ∆ × ( X ⊠ Y ) → X ⊠ Y . Unravelling definitions this means that for all0 ≤ k ≤ n and all x, y ∈ ( X ⊠ Y ) n also(2.1.1) (cid:0) ( α, . . . , α | {z } k +1 times , , . . . , .x, ( β, . . . , β | {z } k +1 times , , . . . , .y ) ∈ ( X ⊠ Y ) n . But indeed, for any 0 ≤ ℓ ≤ k , the previous lemma implies thatsupp ℓ (( α, . . . , α, . . . , .x ) = α (supp ℓ ( x )) , supp ℓ (( β, . . . , β, . . . , .y ) = β (supp ℓ ( y ))and these are disjoint as im α ∩ im β = ∅ . On the other hand, if ℓ > k , then by thesame argumentsupp ℓ (( α, . . . , α, . . . , .x ) = supp ℓ ( x ) , supp ℓ (( α, . . . , α, . . . , .y ) = supp ℓ ( y )which are again disjoint by assumption. Thus, Φ restricts to the desired homotopybetween ir and the identity, finishing the proof of the theorem. (cid:3) Applying the theorem inductively, we in particular see that the inclusion X ⊠ · · · ⊠ X n ֒ → X × · · · X n is a G -global weak equivalence for all X , . . . , X n ∈ E M - G -SSet τ . Specializing to X = · · · = X n =: X we see that X ⊠ n ֒ → X × n isa G -global weak equivalence. However, the right hand side has an additional Σ n -action given by permuting the factors, and the left hand side is preserved by this.The following theorem then refines the above comparison to take this additionalaction into account: Theorem . For any X ∈ E M - G -SSet τ the inclusion X ⊠ n ֒ → X × n isa ( G × Σ n ) -global weak equivalence. Proof.
Let H ⊂ M be any universal subgroup and let ϕ : H → Σ n × G be a weak equivalence. We have to show that the inclusion i induces a weakhomotopy equivalence on ϕ -fixed points. To this end let us write ϕ : H → Σ n forthe projection of ϕ to the first factor. Then it is obviously enough to show that i is a ( G × H )-equivariant homotopy equivalence where H acts on both sides via ϕ and the E M -action and G acts as before. .1. ULTRA-COMMUTATIVITY 117 We now pick an H -equivariant injection u : { , . . . , n } × ω ω where H actson { , . . . , n } via ϕ and on ω as before; this exists since the source is countableand the target is a complete H -set universe. If we write u i := u ( i, –) : ω → ω thenthe u i ’s are elements of M with pairwise disjoint image and the H -equivariance of u translates into the relation u i h = hu h − .i for all i = 1 , . . . , n . Let us now define r : X × n → X ⊠ n as the restriction of n Y i =1 ( u i . –) : X × n → X × n ;note that this indeed lands in X ⊠ n ⊂ X × n as for any n -tuple ( x , . . . , x n ) in theimage already supp( x i ) ∩ supp( x j ) = ∅ for i = j as the images of the u i ’s arepairwise disjoint.It is obvious that r is G -equivariant. But it is also H -equivariant: if ( x , . . . , x n )is any m -simplex of X × n , then(2.1.2) r ( h. ( x , . . . , x n )) = r ( h.x h − . , . . . , h.x h − .n )= (cid:0) u . ( h.x h − . ) , . . . , u n . ( h.x h − .n ) (cid:1) = ( h.u h − . .x h − . , . . . , h.u h − .n .x h − .n )= h. ( u .x , . . . , u n .x n ) = h. (cid:0) r ( x , . . . , x n ) (cid:1) . Thus it only remains to construct ( H × G )-equivariant homotopies ir ≃ id and ri ≃ id. For the first one we observe that we have for each i = 1 , . . . , n a homotopyid ⇒ u i given by the action of ( u i , ∈ ( E M ) and these assemble into a homotopyΦ : id ⇒ ir . This homotopy is obviously G -equivariant, and it is moreover H -equivariant by a similar calculation as in (2 . . G × H )-equivariant homotopy.To finish the proof it is then enough to show that Φ also restricts to a homotopyid ⇒ ri , for which one argues precisely as in the proof of Theorem 2.1.11. (cid:3) Together with Lemma 1.4.46 we immediately conclude:
Corollary . Let f : X → Y be a G -global weak equivalence of tame E M - G -simplicial sets. Then f ⊠ n : X ⊠ n → Y ⊠ n is a ( G × Σ n ) -global weak equiva-lence. (cid:3) The following corollary will be crucial later for establishing model structureson categories of commutative monoids later:
Corollary . Let f : X → Y be a G -global weak equivalence of tame E M - G -simplicial sets, and assume that neither X nor Y contain vertices of emptysupport. Then Sym n ( f ) := f ⊠ n / Σ n is a G -global weak equivalence for all n ≥ . Proof.
The previous corollary asserts that f ⊠ n is a ( G × Σ n )-global weakequivalence. By Corollary 1.4.45 it is therefore enough to show that Σ n acts freelyon both X ⊠ n and Y ⊠ n .We prove the first statement, the other one being analogous. For this it sufficesto show that Σ n acts freely on ( X ⊠ n ) ; but indeed, if ( x , . . . , x n ) is any vertexand i = j , then supp( x i ) ∩ supp( x j ) = ∅ = supp( x j ), so supp( x i ) = supp( x j ) andhence in particular x i = x j .
18 2. COHERENT COMMUTATIVITY
If now σ ∈ Σ n is non-trivial, then we find i with σ − ( i ) = i , so ( x , . . . , x n )and σ. ( x , . . . , x n ) = ( x σ − (1) , . . . , x σ − ( n ) ) differ in the i -th component. (cid:3) The box product as an operadic product.
Finally, we come to an al-ternative description of the box product of tame E M -simplicial sets analogousto [ SS20 , Proposition A.17]; this construction (as well as the existence of the iso-morphism in Theorem 2.1.17 below) was suggested to me by Stefan Schwede.
Construction . Let X , . . . , X n be E M -simplicial sets. We consider(2.1.3) E Inj( n × ω, ω ) × ( E M ) n ( X × · · · X n ) , i.e. the quotient of E Inj( n × ω, ω ) × ( X × · · · X n ) by the equivalence relationgenerated on m -simplices by( f , . . . , f m ; ( u (1)0 , . . . , u (1) m ) .x , . . . ( u ( n )0 , . . . , u ( n ) m ) .x n , ) ∼ ( f u • , · · · , f m u • m ; x , . . . , x n )for all injections f , . . . , f m : n × ω ω , all u ( j ) i ∈ M ( i = 0 , . . . , m , j = 1 , . . . , n ),and ( x , . . . , x n ) ∈ ( X ×· · · X n ) m ; here we write u • i : n × ω → n × ω for the injectionwith u • i ( j, t ) = ( j, u ( j ) i ( t )).The simplicial set (2 . .
3) has a natural E M -action given by postcomposition.Moreover one easily checks, that for all f i : X i → X ′ i the map E Inj( n × ω, ω ) × ( f ×· · · f n ) descends to E Inj( n × ω, ω ) × ( E M ) n ( X × · · · X n ) → E Inj( n × ω, ω ) × ( E M ) n ( X ′ × · · · × X ′ n ), and that this yields a functor ( E M -SSet ) n → E M -SSet .Finally, we have a natural E M -equivariant mapΦ : E Inj( n × ω, ω ) × ( E M ) n ( X × · · · × X n ) → X × · · · × X n given on m -simplices by( f , . . . , f n ; x , . . . , x n ) (( f ι , . . . , f m ι ) .x , . . . , ( f ι n , . . . , f m ι n ) .x n )where ι k : ω → n × ω is defined via ι k ( t ) = ( k, t ). Theorem . For any X , . . . , X n ∈ E M -SSet τ the map Φ restricts toan isomorphism E Inj( n × ω, ω ) × ( E M ) n ( X × · · · × X n ) → X ⊠ · · · ⊠ X n . For the proof we will need:
Lemma . Let ( x , . . . , x n ) ∈ ( X × · · · × X n ) m , f , . . . , f m ; g , . . . , g m ∈ Inj( n × ω, ω ) , and assume f k and g k agree on S ni =1 { i } × supp k ( x i ) for all k . Then [ f , . . . , f m ; x , . . . , x n ] = [ g , . . . , g m ; x , . . . , x n ] in E Inj( n × ω, ω ) × ( E M ) n ( X ×· · · × X n ) ,. Proof.
By induction we may assume that there exists an i ′ such that f i = g i for all i = i ′ , and by symmetry we may assume that i ′ = 0. We now consider ϕ : Inj( n × ω, ω ) → (cid:0) E Inj( n × ω, ω ) × ( E M ) n ( X × · · · × X n ) (cid:1) m h [ h, f , . . . , f m ; x , . . . , x n ] = [ h, g , . . . , g m ; x , . . . , x n ] . We claim that this factors over Inj( n × ω, ω ) / ( M supp ( x ) ×· · ·×M supp ( x n ) ): indeed,if u ( i ) ∈ M supp ( x i ) [ h, f , . . . , f m ; x , . . . , x n ] = [ h, f , . . . , f m , ( u (1) , , . . . , .x , . . . , ( u ( n ) , , . . . , .x n ]= [ hu • , f , . . . , f m ; x , . . . , x n ] .1. ULTRA-COMMUTATIVITY 119 by definition, i.e. ϕ is compatible with generating relations and hence factors overthe quotient. But [ f ] = [ g ] in Inj( n × ω, ω ) / ( M supp ( x ) × · · · × M supp ( x n ) )by [ SS20 , Lemma A.5], which completes the proof the lemma. (cid:3)
Proof of Theorem 2.1.17.
Lemma 2.1.12 readily implies that the image ofΦ is contained in X ⊠ · · · ⊠ X n . Let us now prove the other inclusion: if ( x , . . . , x n )is any m -simplex of X ⊠ · · · ⊠ X n , then we pick for each k an injection f k : n × ω → ω with f k ( i, t ) = t for all t ∈ supp k ( x i ); this is possible as the sets supp k ( x i ) arepairwise disjoint and finite for fixed k and varying i . It is then easy to check thatΦ[ f , . . . , f m ; x , . . . , x n ] = ( x , . . . , x n ).For injectivity we pick f , . . . , f m ; g , . . . , g m ∈ Inj( n × ω, ω ) and ( x , . . . , x n ) , ( y , . . . , y n ) ∈ X ×· · ·× X n with Φ[ f , . . . , f m ; x , . . . , x n ] = Φ[ g , . . . , g m ; y , . . . , y n ]. Claim. If f k = g k for all k = 0 , . . . , m , then also x j = y j for all j = 1 , . . . , n . Proof.
We have ( f ι j , . . . , f m ι j ) .x j = ( g ι j , . . . , g m ι j ) .y j = ( f ι j , . . . , f m ι j ) .y j .The claim follows as ( f ι j , . . . , f m ι j ) . – : X × · · · × X n is injective as compositionof the injections i ( f ι j ) . – , . . . , i m ( f m ι j ) . –. △ In the general case we observe that for each 1 ≤ j ≤ n and 0 ≤ k ≤ m in partic-ular supp k (( f k ι j ) .x k ) = supp k (( g k ι j ) .y k ), so ( f k ι j )(supp k x k ) = ( g k ι j )(supp k ( y k ))by Lemma 2.1.12. By injectivity of f k ι j and g k ι j it follows that there exists a(unique) bijection σ ( j ) k : supp k ( x k ) → supp k ( y k ) with g k ι j σ ( j ) k = f k ι j | supp k ( x k ) .We now pick an extension of σ ( j ) k to an injection s ( j ) k ∈ M . Letting j vary, weconclude from the above definition that g k s • k and f k agree on S nj =1 { j } × supp k ( x k ).Letting k vary, Lemma 2.1.18 therefore shows[ f , . . . , f m ; x , . . . , x n ] = [ g s • , . . . , g n s • k ; x , . . . , x n ]= [ g , . . . , g n ; ( s (1)0 , . . . , s (1) m ) .x , . . . , ( s ( n )0 , . . . , s ( n ) m ) .x n ] . In particular,Φ[ g , . . . , g m ; y , . . . , y n ] = Φ[ f , . . . , f m ; x , . . . , x n ]= Φ[ g , . . . , g n ; ( s (1)0 , . . . , s (1) m ) .x , . . . , ( s ( n )0 , . . . , s ( n ) m ) .x n ];the claim therefore implies that ( s ( j )0 , . . . , s ( j ) m ) .x j = y j , finishing the proof. (cid:3) Corollary . The box product of tame E M -simplicial sets preservestensors and small colimits in each variable. Proof.
By the above it suffices to check this for E Inj( × ω, ω ) × ( E M ) (– × –),which is obvious. (cid:3) Purely by abstract nonsense, the sym-metric monoidal structure on I extends uniquely to a simplicially enriched symmet-ric monoidal structure on I . In particular, we get again a Day convolution producton I -SSet characterized by I ( A, –) ⊠ I ( B, –) = I ( A ∐ B, –). Proposition . There exists a unique enhancement of ev ω : I -SSet → E M -SSet τ to a strong symmetric monoidal functor such that for all finite sets A, B the structure isomorphism (2.1.4) (cid:0) I ( A, –) ⊠ I ( B, –) (cid:1) ( ω ) = I ( A ∐ B, –)( ω ) → I ( A, –)( ω ) ⊠ I ( B, –)( ω ) is induced by the map I ( A ∐ B, –) → I ( A, –) × I ( B, –) given by restriction.
20 2. COHERENT COMMUTATIVITY
Proof.
As seen in Example 2.1.8, the restrictions induce an isomorphism E Inj( A ∐ B, ω ) ∼ = E Inj(
A, ω ) ⊠ E Inj(
B, ω ). Using the canonical isomorphisms I ( C, –)( ω ) ∼ = E Inj(
C, ω ) for C ∈ I , we therefore conclude that (2 . .
4) is well-defined and an isomorphism.As ev ω is cocontinuous and preserves tensors, the universal property of enrichedpresheaves now easily implies that (2 . .
4) extends to a unique simplicial naturalisomorphism ψ : ev ω (– ⊠ –) ⇒ ev ω ⊠ ev ω . On the other hand, there is a uniqueisomorphism ι : I ( ∅ , –) → ∗ as both sides are terminal. It only remains to showthat ψ and ι endow ev ω with the structure of a strong symmetric monoidal functor.For this we simply observe that it again suffices to check the coherence condi-tions on representables, where this is a trivial calculation. (cid:3) We can define the box product of G - I -simplicial sets by pulling through the G -action again. The above then induces a (simplicial) strong symmetric monoidalstructure on ev ω : G - I -SSet → E M - G -SSet τ . Corollary . The box product of G - I -simplicial sets is homotopical ineach variable. Proof.
Let f : X → X ′ and g : Y → Y ′ be G -global weak equivalences. Wehave to show that f ⊠ g is a G -global weak equivalence, i.e. that ( f ⊠ g )( ω ) is a G -global weak equivalence of E M - G -simplicial sets. But by the previous propositionthis is conjugate to f ( ω ) ⊠ g ( ω ), and by assumption both f ( ω ) and g ( ω ) are G -globalweak equivalences. The claim therefore follows from Theorem 2.1.11. (cid:3) Corollary . Let f : X → Y be a G -global weak equivalence of G - I -simplicial sets such that X and Y are cofibrant in the positive G -global modelstructure. Then Sym n f := f ⊠ n / Σ n is a G -global weak equivalence for all n ≥ . Proof.
As in the previous corollary, we see that f ⊠ n ( ω ) agrees with f ( ω ) ⊠ n up to conjugation by ( G × Σ n )-equivariant isomorphisms (the Σ n -equivariance usesthat we have a strong symmetric monoidal structure). As ev ω is a left adjoint,it commutes with quotients, so we conclude that also (Sym n f )( ω ) agrees withSym n ( f ( ω )) up to conjugation.By Corollary 2.1.15 we are therefore reduced to showing that Z ( ω ) has no ele-ments of empty support for any Z cofibrant in the positive G -global model struc-ture. This follows from Lemma 1.4.39, as ev ω : G - I -SSet → E M - G -SSet is leftQuillen. (cid:3) Definition . We write G -UCom := CMon( G - I -SSet ) and call itsobjects G -ultra-commutative monoids .2.1.3.1. A reminder on model structures for commutative monoids.
We wantto construct a suitable G -global model structure on G -UCom , for which we needto recall the general machinery of [ Whi17 ] and [
GG16 ]. Definition . A model category C equipped with a closed symmetricmonoidal structure is called a symmetric monoidal model category if the followingtwo conditions are satisfied:(1) ( Unit Axiom ) If X ∈ C is cofibrant, then I ⊗ X → ⊗ X is a weakequivalence for some (hence any) cofibrant replacement I ∼ −→ of theunit. .1. ULTRA-COMMUTATIVITY 121 (2) ( Pushout Product Axiom ) If i : X → X ′ and j : Y → Y ′ are cofibrations,then so is the pushout product map i (cid:3) j : ( X ⊗ Y ′ ) ∐ X ⊗ Y ( X ′ ⊗ Y ) → X ′ ⊗ Y ′ . Moreover, if at least one of i and j is acyclic, then so is i (cid:3) j .It is a well-known fact that for cofibrantly generated C it suffices to checkthe pushout product axiom on some chosen generating (acyclic) cofibrations, seee.g [ SS00 , Lemma 3.5-(1)].
Definition . A symmetric monoidal model category C satisfies the Monoid Axiom if every transfinite composition of pushouts of maps of the form X ⊗ j with X ∈ C and j an acyclic cofibration is a weak equivalence again.If C is again cofibrantly generated, then for verifying the Monoid Axiom it isenough to restrict to the case that j belongs to a chosen set J of generating acycliccofibrations, i.e. the above condition can be reformulated as saying that any relative( C ⊗ J )-cell complex is a weak equivalence, see [ SS00 , Lemma 3.5-(2)].It is a classical result of Schwede and Shipley [
SS00 , Theorem 4.1-(3)] thatfor a combinatorial symmetric monoidal model category C satisfying the MonoidAxiom the transferred model structure on the category Mon( C ) of not necessarilycommutative monoids exists, i.e. Mon( C ) can be equipped with a model structurein which the fibrations and weak equivalences are created in C . We will need aversion of this for commutative monoids due to White [ Whi17 ]. This relies on thefollowing additional notion:
Construction . Let C be a symmetric monoidal category with finitecolimits. As usual in the context of model structures on commutative monoids, wewill suppress the associativity isomorphisms to keep the notation tractable. Thereader uncomfortable with this may rest assured that we will only ever apply thisto box products, which have a preferred unbiased n -ary tensor product anyhow.For n ≥ C n denote the n -cube, i.e. the poset of subsets of { , . . . , n } . If f : X → Y is any morphism in C , then we have a functor K nn ( f ) : C n → C sending I ⊂ { , . . . , n } to Z ⊗ · · · ⊗ Z n where Z i = Y i if i ∈ I , and Z i = X i otherwise; thestructure maps are induced by the obvious tensor products of f and the respectiveidentities.We write Q nn − ( f ) for the colimit of the subdiagram obtained by removing theterminal vertex of C n . This has a natural Σ n -action obtained by the Σ n -action onthe punctured n -cube and the symmetry isomorphisms of the tensor product on C .The natural map f (cid:3) n : Q nn − ( f ) → Y ⊗ n induced by the remaining structure mapsof K nn is then equivariant with respect to the Σ n -action on the target via permutingthe tensor factors. Remark . Note that f (cid:3) n would be usually used to denote the iteratedpushout product of f with itself, which is indeed canonically and Σ n -equivariantlyconjugate to the above, see e.g. [ GG16 , Section 3]. However, we will only careabout the above description except in the proof of Corollary 2.1.38.
Definition . A symmetric monoidal model category C satisfies the Strong Commutative Monoid Axiom if i (cid:3) n / Σ n is a a cofibration for all n ≥ i in C , acyclic if i is.Although this is much harder to prove than the analogous statements for theprevious axioms we still have:
22 2. COHERENT COMMUTATIVITY
Lemma . Let C be a cofibrantly generated symmetric monoidal modelcategory, let I be a set of generating cofibrations, and let J be a set of generat-ing acyclic cofibrations. Then C satisfies the Strong Commutative Monoid Axiomprovided that i (cid:3) n / Σ n is a cofibration for any i ∈ I and that j (cid:3) n / Σ n is an acycliccofibration for all j ∈ J . Proof.
See [
Whi17 , proof of Lemma A.1] or [
GG16 , Corollary 9]. (cid:3)
In many practical situations the following lemma due to Gorchinskiy and Gulet-ski˘ı further simplifies the verification of the Strong Commutative Monoid Axiom:
Lemma . Let C be a cofibrantly generated model category, let I be a setof generating cofibrations, and let J be a set of generating acyclic cofibrations suchthat all maps in J have cofibrant source . Then C satisfies the Strong CommutativeMonoid Axiom provided that i (cid:3) n / Σ n is a cofibration for all i ∈ I and that Sym n j = j ⊗ n / Σ n is a weak equivalence for all j ∈ J . Proof.
By the argument from the proof of the previous lemma, i.e. by [
GG16 ,Corollary 9], i (cid:3) n / Σ n is a cofibration for all (not necessarily generating) cofibrations i . It remains to show that the cofibration j (cid:3) n / Σ n is acyclic for each j ∈ J . Butas j is by assumption a map of cofibrant objects, this is a special case of [ GG16 ,Corollary 23], also see [
GG16 , Corollary 10]. (cid:3)
The forgetful functor CMon( C ) → C has a left adjoint P given on objects by X a n ≥ X ⊗ n / Σ n = a n ≥ Sym n X and analogously on morphisms; the monoid structure is given by concatenation. Theorem . Let C be a combinatorial symmetric monoidal modelcategory satisfying both the Monoid Axiom and the Strong Commutative MonoidAxiom. Then there exists a unique model structure on CMon( C ) in which a mapis a weak equivalence or fibration if and only if it so in C . This model structure isagain combinatorial, and when I and J are generating cofibrations and generatingacyclic cofibrations of C , then P I and P J are sets of generating cofibrations andgenerating acyclic cofibrations, respectively, for CMon( C ) . Proof.
See [
Whi17 , Theorem 3.2] and the discussion after it. (cid:3)
Theorem . In the situation of the previous theorem,
CMon( C ) is left proper provided that the following additional conditions are satisfied: (1) C is left proper and filtered colimits in it are homotopical. Moreover, thereexists a set of generating cofibrations with cofibrant sources . (2) If X ∈ C and i is any cofibration, then pushouts in C along X ⊗ i arehomotopy pushouts. Moreover, if X is cofibrant, then X ⊗ – is homotopical. Proof.
This is a special case of [
Whi17 , Theorem 4.17], also see [
Whi17 ,discussion after Definition 4.15] and [
BB17 , discussion after Definition 2.4]. (cid:3) .1. ULTRA-COMMUTATIVITY 123
Construction of the model structure.
We can now prove:
Theorem . There is a unique model structure on G -UCom in which amap is a weak equivalence or fibration if and only if it is so in the positive G -globalmodel structure on G - I -SSet .We call this the positive G -global model structure . It is proper, simplicial, andcombinatorial with generating cofibrations (say, as maps in G - I -SSet ) a n ≥ (cid:0) I ( n × A, –) × G n (cid:1) / (Σ n ≀ H ) × ( ∂ ∆ m ֒ → ∆ m ) × n , where m ≥ , H runs through all finite groups, and A through finite faithful non-empty H -sets. Moreover, filtered colimits in this model category are homotopical.Finally, the simplicial adjunction P : G - I -SSet positive G -global ⇄ G -UCom positive G -global : forget is a Quillen adjunction. Proof.
Let us first establish the model structure, for which it suffices to verifythe assumptions of Theorem 2.1.31.In order to verify the Pushout Product for cofibrations, we may restrict to thestandard generating cofibrations. We therefore let H , H be finite groups, we let A i ( i = 1 ,
2) be a finite faithful non-empty H i -set, we let ϕ i : H i → G ( i = 1 , n , n ≥
0. We consider I ( A , –) × G with H acting from the right on the first factor via its action on A and on the secondfactor via ϕ ; similarly we equip I ( A , –) × G with a right H -action. Then (cid:0) I ( A , –) × G × ( ∂ ∆ n ֒ → ∆ n ) (cid:1) (cid:3) (cid:0) I ( A , –) × G × ( ∂ ∆ n ֒ → ∆ n ) (cid:1) agrees up to conjugation by isomorphisms with I ( A ∐ A , –) × (cid:0) ( G × ∂ ∆ n ֒ → G × ∆ n ) (cid:3) ( G × ∂ ∆ n ֒ → G × ∆ n ) (cid:1) . This isomorphism is equivariant in H , H , and G , if we let G act in the obviousway, H via its action on A and its action on the first G -factor, and H via itsaction on A and its action on the second G -factor. All the H -actions commutewith all the H -actions, so that they assemble into a H × H -action. As ⊠ preservescolimits in each variable, we conclude that (cid:0) ( I ( A , –) × ϕ G ) × ( ∂ ∆ n ֒ → ∆ n ) (cid:3) (cid:0) ( I ( A , –) × ϕ G ) × ( ∂ ∆ n ֒ → ∆ n ) agrees up to conjugation by isomorphisms with (cid:0) I ( A ∐ A , –) × (cid:0) ( G × ∂ ∆ n ֒ → G × ∆ n ) (cid:3) ( G × ∂ ∆ n ֒ → G × ∆ n ) (cid:1)(cid:1) / ( H × H ) . The ( H × H )-set A ∐ A is obviously faithful, and ( G × ∂ ∆ n ֒ → G × ∆ n ) (cid:3) ( G × ∂ ∆ n ֒ → G × ∆ n ) is injective by the Pushout Product Axiom for SSet , moreover G clearly acts freely on the target. We claim that this already implies that theabove is a positive G -global cofibration. Indeed, for any faithful H -set A = ∅ theadjunction I ( A, –) × H – : ( G × H op )-SSet ⇄ G - I -SSet : ev A is a Quillen adjunction with respect to the G H op ,G -equivariant model structure onthe source and the G -global positive level model structure on the target, hence inparticular with respect to the G -global positive model structure. The claim followsas the cofibrations on the left hand side are precisely the injections i such that G acts freely outside the image of i .
24 2. COHERENT COMMUTATIVITY
This proves the Pushout Product Axiom for cofibrations. For the part aboutacyclic cofibrations we will use:
Claim.
Let i : A → B be an injective cofibration in G - I -SSet and let X bearbitrary. Then any pushout along X ⊠ i is a homotopy pushout. Proof.
The functor ev ω reflects homotopy pushouts and it it preserves ordi-nary pushouts. It is therefore enough to show that any pushout along ( X ⊠ i )( ω ) in E M - G -SSet is a homotopy pushout. But ev ω is strong symmetric monoidal, sothis is conjugate to X ⊠ i ( ω ) which is evidently an injective cofibration. The claimfollows immediately. △ If now i : X → Y is any cofibration, and i : X → Y is any acyclic cofibra-tion, then we consider the commutative diagram X ⊠ X Y ⊠ X X ⊠ Y P Y ⊠ Y . p X ⊠ i i ⊠ X Y ⊠ i i ⊠ Y i (cid:3) i By the above the top square is a homotopy pushout, and the left hand arrow as wellas the rightmost arrow are weak equivalences as ⊠ is homotopical (Corollary 2.1.21).We conclude that Y ⊠ X → P is a weak equivalence and hence so is i (cid:3) i by2-out-of-3. Since we already know that it is a cofibration, it is therefore an acycliccofibration. This completes the verification of the Pushout Product Axiom.The Unit Axiom is immediate because ⊠ is fully homotopical by Corollary 2.1.21.Let us now verify the Monoid Axiom. If j is any acyclic cofibration, then it isin particular an injective cofibration. If now X is any object, then pushouts along X ⊠ j are therefore homotopy pushouts by the above claim. However, X ⊠ j isalso a weak equivalence because ⊠ is homotopical, so any pushout of it is a weakequivalence as desired.Next, we consider the Strong Commutative Monoid Axiom. We let H be afinite group, A a finite faithful non-empty H -set, and m, n ≥
0. Then(2.1.5) (cid:0) ( I ( A, –) × G ) × ( ∂ ∆ m ֒ → ∆ m ) (cid:1) (cid:3) n agrees up to conjugation by isomorphisms with(2.1.6) I ( n × A, –) × ( G × ∂ ∆ m ֒ → G × ∆ m ) (cid:3) n . Now let ϕ : H → G be any group homomorphism. There are n commuting H -actions on (2 . . i -th one of which is given by acting on the i -th (cid:3) -summandin the obvious way. We similarly have n commuting H -actions on n × A with the i -thaction given by acting in the prescribed way on the i -th copy of A and trivially on allother copies, and we have n further commuting H -actions on ( G × ∂ ∆ m ֒ → ∆ m ) (cid:3) n analogously to the above. By taking the respective diagonals we get n commuting H -actions on (2 . . n acts on n × A via its tautological action on n and on the (cid:3) -powers via permuting the factors. Again taking the diagonal for (2 . .
6) we getΣ n -actions compatible with the above identification. .1. ULTRA-COMMUTATIVITY 125 We now observe that the Σ n -actions and the H -actions assemble into (Σ n ≀ H )-actions on both (2 . .
5) and (2 . .
6) and we altogether conclude that (cid:0) ( I ( A, –) × ϕ G ) × ( ∂ ∆ m ֒ → ∆ m ) (cid:1) (cid:3) n / Σ n agrees up to conjugation by isomorphisms with(2.1.7) (cid:0) I ( n × A, –) × ( G × ∂ ∆ m ֒ → ∆ m ) (cid:3) n (cid:1) / (Σ n ≀ H )But the (Σ n ≀ H )-action on n × A is faithful: if ( σ ; h , . . . , h n ) acts trivially on n × A , then each h i has to act trivially on A , so h i = 1 for all i . But then if a ∈ A is arbitrary (here we used that A = ∅ ), then ( σ ; h , . . . , h n ) = σ sends ( i, a ) to( σ ( i ) , a ), so also σ = 1. Thus, (2 . .
7) is a cofibration by the same argument as inthe verification of the Pushout Product Axiom.For the Strong Commutative Monoid Axiom for acyclic cofibrations, we observethat the positive G -global model structure on G - I -SSet is combinatorial and thatthe standard generating cofibrations have cofibrant sources. By [ Bar10 , Corol-lary 2.7] we may conclude that there exists a set of generating acyclic cofibrations J with cofibrant sources. By Lemma 2.1.30 it therefore suffices that for each j ∈ J and n ≥ n j is a weak equivalence. But as the source (and hence alsothe target) of j was assumed to be cofibrant in the positive flat model structure,this is an instance of Corollary 2.1.22.This completes the verification of the assumptions of Theorem 2.1.31; we con-clude that the positive G -global model structure on G -UCom exists and is com-binatorial with generating cofibrations P I and generating acyclic cofibrations P J ,where I and J are sets of generating cofibrations and generating acyclic cofibra-tions for G - I -SSet . If we take I to be the standard generating cofibrations, then acalculation analogous to the verification of the Strong Commutative Monoid Axiomidentifies P I with the proposed generating cofibrations.The model structure constructed this way is right proper because it is trans-ferred from a right proper model structure, and it is also left proper by an imme-diate application of Theorem 2.1.32, all of whose assumptions have been verifiedabove. Moreover, filtered colimits in it are homotopical, as the forgetful functor to G - I -SSet creates both weak equivalences as well as filtered colimits.Moreover, CMon( G - I -SSet ) has an evident simplicial enrichment, and it iscotensored over SSet with cotensors formed in G - I -SSet . For each fixed K , (–) K preserves limits and sufficiently highly filtered colimits (since these are created bythe forgetful functor), so it admits a left adjoint by the Special Adjoint FunctorTheorem, i.e. CMon( G - I -SSet ) is also tensored over SSet . Finally, the forgetfulfunctor has an obvious enrichment with respect to which it preserves cotensors, sothat P ⊣ forget becomes a simplicial adjunction. We may therefore conclude asbefore that the positive G -global model structure on G -UCom is simplicial.Finally, the forgetful functor is right Quillen by design, so that P ⊣ forget is asimplicial Quillen adjunction. (cid:3) G -parsummable simplicial sets. Wenow want to lift our comparison between E M - G -SSet τ and G - I -SSet : Theorem . There exist a (unique) model structure on G -ParSumSSet in which a map is a weak equivalence or fibration if and only if it so in the positive G -global model structure on E M - G -SSet τ .
26 2. COHERENT COMMUTATIVITY
We call this the positive G -global model structure . It is proper, simplicial, andcombinatorial with generating cofibrations (say, as maps in G - E M -SSet τ ) a n ≥ (cid:0) E Inj( n × A, ω ) × G n (cid:1) / (Σ n ≀ H ) × ( ∂ ∆ m ֒ → ∆ m ) × n where m ≥ , H runs through all finite groups, and A is a finite faithful non-empty H -set. Moreover, filtered colimits in this model category are homotopical.Finally, the simplicial adjunction P : ( E M - G -SSet τ ) positive G -global ⇄ G -ParSumSSet positive G -global : forget is a Quillen adjunction. Proof.
We again verify the assumptions of Theorems 2.1.31 and 2.1.32.For the Pushout Product Axiom it is enough to check this on generating(acyclic) cofibrations. As these are given by applying ev ω to the generating (acyclic)cofibrations of G - I -SSet and since ev ω is strong symmetric monoidal and cocontin-uous, this is a formal consequence of the Pushout Product Axiom for G - I -SSet .Analogously, the Strong Commutative Monoid Axiom follows from the one for G - I -SSet .The Unit Axiom is again automatic as ⊠ is homotopical (Theorem 2.1.11).Next, we observe that X ⊠ – preserves injective cofibrations, which immediatelyimplies that pushouts along X ⊠ i are homotopy pushouts for any positive G -globalcofibration i . Thus, the Monoid Axiom holds by the same argument as before.We conclude that the model structure exists, is proper, and that filtered colimitsin it are homotopical. Moreover, it is combinatorial with generating cofibrations P I for any set I of generating cofibrations of E M - G -SSet τ . Again using that ev ω is cocontinuous and strong symmetric monoidal, the calculation from the previoustheorem yields the above description of P I .Finally, one argues as in the previous theorem to establish that the modelstructure is simplicial (with cotensoring created in E M - G -SSet τ ). (cid:3) In order to compare this model category to the one from the previous sec-tion, we first abserve that there is again by abstract nonsense a preferred wayto make the right adjoint (–) • : E M -SSet τ into a lax monoidal functor, so thatwe get an induced adjunction ev ω : G -UCom ⇄ G -ParSumSSet : (–) • . ByLemma 1.4.25 the right adjoint is fully faithful with essential image precisely those G -ultracommutative monoids whose underlying G - I -simplicial sets are flat. Warning . As opposed to the situation for I -simplicial sets, I do notknow whether the above makes (–) • into a strong symmetric monoidal functor,i.e. whether the canonical maps X • ⊠ Y • → ( X ⊠ Y ) • are isomorphisms. As (–) • isfully faithful with essential image precisely the flat I -simplicial sets, this is equiva-lent to the question whether the box product of I -simplicial sets preserves flatness. Corollary . The simplicial adjunction ev ω : G -UCom ⇄ G -ParSumSSet : (–) • is a Quillen equivalence. Proof.
While this follows from the corresponding result for the underlyingadjunction G - I -SSet ⇄ E M - G -SSet τ by [ Whi17 , Theorem 4.19], there is aneasy direct argument available: by construction, (–) • preserves weak equivalences .1. ULTRA-COMMUTATIVITY 127 and fibrations and (ev ω ) creates weak equivalences, so it suffices that the counit X • ( ω ) → X is a weak equivalence for every (fibrant) X . But this is given by thecounit of the underlying adjunction, so it is in fact an isomorphism for all X . (cid:3) For later use we record:
Lemma . Let i : A → B and j : C → D be injective cofibrations of tame E M -simplicial sets. Then i (cid:3) j is an injective cofibration. Proof.
It is clear that B ⊠ j : B ⊠ C → B ⊠ D and i ⊠ D are injective.Now assume ( a, d ) ∈ ( A ⊠ D ) n and ( b, c ) ∈ ( B ⊠ C ) n have the same image in( B ⊠ D ) n . Then by definition b = i ( a ) and d = j ( c ); so to show that ( b, c ) and( a, d ) represent the same element in the pushout ( A ⊠ D ) ∐ A ⊠ C ( B ⊠ C ) it sufficesthat ( a, c ) ∈ ( A ⊠ C ) n , i.e. supp k ( a ) ∩ supp k ( c ) = ∅ for all 0 ≤ k ≤ n . But indeed,as i is injective and E M -equivariant, supp k ( a ) = supp k ( i ( a )) = supp k ( b ), which isdisjoint from supp k ( c ) as ( b, c ) ∈ ( B ⊠ C ) n . (cid:3) Corollary . Let i : A → B be an injective cofibration of tame E M -simplicial sets, and let n ≥ . Then i (cid:3) n and i (cid:3) n / Σ n are injective cofibrations. Proof.
Identifying i (cid:3) n with the iterated pushout product, the first claim fol-lows by applying the previous lemma inductively. The second statement follows asquotients by group actions in Set preserve injections. (cid:3) I vs. I and M vs. E M . We will now establish analogues of theabove model structures for commutative monoids in G - I -SSet and M - G -SSet ,and show that these are equivalent in the evident way to the models considered sofar. This will in particular allow us in the next subsection to compare our notionof ultra-commutative monoids to the one considered by Schwede.We begin with the following observation that is proven analogously to Propo-sition 2.1.20: Lemma . There is a unique way to make
I × I – : I -SSet → I -SSet intoa strong symmetric monoidal functor such that the structure isomorphism (cid:0) I × I I ( A, –) (cid:1) ⊠ (cid:0) I × I I ( B, –) (cid:1) → I × I (cid:0) I ( A, –) ⊠ I ( B, –) (cid:1) is the identity for all A, B ∈ I . (cid:3) We can now deduce the following G -global analogue of [ SS20 , Theorem 1.1]:
Theorem . The box product on G - I -SSet is fully homotopical in G -global weak equivalences. Proof.
By Theorem 1.4.27 the functor
I × I – detects G -global weak equiva-lences (without any need to derive). The claim therefore follows from the previouslemma together with Corollary 2.1.21. (cid:3) Theorem . There is a unique model structure on
CMon( G - I -SSet ) inwhich a map is a weak equivalence or fibration if and only if it is so in the positive G -global model structure on G - I -SSet . We call this the positive G -global modelstructure . It is left proper, simplicial, combinatorial, and filtered colimits in it arehomotopical. Finally, the simplicial adjunction (2.1.8) I × I – : G - I -SSet ⇄ G - I -SSet = G -UCom : forget is a Quillen equivalence in which both adjoints are fully homotopical.
28 2. COHERENT COMMUTATIVITY
Here we again used that forget acquires a natural lax symmetric monoidalstructure from the strong symmetric monoidal structure on
I × I –. Proof.
We will once again verify the assumptions of Theorem 2.1.31: thePushout Product Axiom for cofibrations is verified analogously to Theorem 2.1.33,and for acyclic cofibrations it then follows from the one for G - I -SSet as I × I – isleft Quillen, symmetric monoidal, and reflects weak equivalences. Analogously, oneverifies the Strong Commutative Monoid Axiom and the Monoid Axiom. Finally,the Unit Axiom is again automatic by Theorem 2.1.40.We therefore conclude as before that the model structure exists, is right proper,simplicial, combinatorial, and that filtered colimits in it are homotopical.As the forgetful functors create fibrations and weak equivalences we concludefrom Theorem 1.4.31 that (2 . .
8) is a Quillen adjunction. As both adjoints in
I × I – : G - I -SSet ⇄ G - I -SSet are fully homotopical by Theorem 1.4.27 andProposition, we conclude that the ordinary unit and counit already represent thederived unit and counit and that they are fully homotopical. From this one easilydeduces that both adjoints in (2 . .
8) are fully homotopical and that unit and counitare weak equivalences, so (2 . .
8) is in particular a Quillen adjunction.Finally, the left Quillen functor
I × I – : CMon( G - I -SSet ) → G -UCom cre-ates weak equivalences, so left properness of CMon( G - I -SSet ) follows from leftproperness of G -UCom as before. (cid:3) Finally, let us turn to the box product on M -SSet : Lemma . For any
X, Y ∈ M -SSet the composition E M × M ( X ⊠ Y ) ֒ → E M × M ( X × Y ) pr , pr −−−−→ ( E M × M X ) × ( E M × M Y ) factors through an isomorphism E M × M ( X ⊠ Y ) → ( E M × M X ) ⊠ ( E M × M Y ) .Together with the unique maps E M × M ∗ → ∗ , the inverses of these isomorphismsmake E M × M – into a strong symmetric monoidal functor. Proof.
Let
X, Y be E M -simplicial sets. Then Lemma 2.1.6 implies thatthe box product of their underlying M -simplicial sets is a subcomplex of theirbox product as E M -simplicial sets, i.e. the inclusions define a natural transforma-tion forget(– ⊠ –) ⇒ forget(–) ⊠ forget(–). As the symmetric monoidal structureisomorphisms of both M -SSet and E M -SSet are defined in terms of the onefor the cartesian monoidal structure, it is clear that this makes forget into a laxmonoidal functor. One easily checks from the definition that the resulting oplaxstructure on E M × M – consists of precisely the above maps E M × M ( X ⊠ Y ) → ( E M× M X ) ⊠ ( E M× M Y ), so it only remains to show that these are isomorphisms.But indeed, the box product of M -simplicial sets is cocontinuous in each vari-able by [ SS20 , Corollary 2.17] and it obviously preserves tensors in each variable.The same holds for the box product of E M -simplicial sets by Theorem 2.1.19. As E M × M – is a simplicial left adjoint, we therefore conclude that the set of all pairs( X, Y ) such that the above comparison map is an isomorphism is closed undertensoring and small colimits in each variable. By Theorem 1.4.8 we are thereforereduced to verifying the claim for X = Inj( A, ω ) and Y = Inj( B, ω ) for some finitesets
A, B . This is then a straightforward calculation using Corollary 1.4.19 togetherwith Examples 2.1.3 and 2.1.8. (cid:3)
Together with Theorem 1.4.22 we conclude as before: .1. ULTRA-COMMUTATIVITY 129
Corollary . The box product on M - G -SSet τ is homotopical in G -global weak equivalences. (cid:3) Theorem . There is a unique model structure on
CMon( M - G -SSet τ ) in which a map is a weak equivalence or fibration if and only if it is so in the positive G -global model structure on M - G -SSet τ . We call this the positive G -global modelstructure . It is left proper, simplicial, combinatorial, and filtered colimits in it arehomotopical. Finally, the simplicial adjunctions (2.1.9) E M × M – : CMon( M - G -SSet τ ) ⇄ CMon( E M - G -SSet τ ) : forget and (2.1.10) ev ω : CMon( G - I -SSet ) ⇄ CMon( M - G -SSet τ ) : (–) • are Quillen equivalences. Proof.
Let us verify the assumptions of Theorem 2.1.31 again.For the Pushout Product Axiom, it suffices to check this on generating cofibra-tions and generating acyclic cofibrations. As the positive G -global model structureon M - G -SSet τ is transferred from G - I -SSet and since ev ω is symmetric monoidal(Remark 2.1.4), it is therefore implied by the pushout product axiom for the latter.Analogously, one deduces the Strong Commutative Monoid Axiom.The Unit Axiom is immediate from the previous corollary, and the MonoidAxiom follows from the one for E M - G -SSet τ (Theorem 2.1.34) as E M × M – isleft Quillen, monoidal, and creates weak equivalences.We therefore conclude that the model structure exists and that it is combinato-rial. Again right properness follows as the forgetful functor creates fibrations, weakequivalences and limits. Similarly, one shows that the resulting model structure issimplicial with respect to the obvious enrichment.The forgetful functor CMon( E M - G -SSet τ ) → CMon( M - G -SSet τ ) preservesweak equivalences and fibrations as they are created in the underlying categories,so it is in particular right Quillen. Similarly, E M × M – is fully homotopical.By Theorem 1.4.22, the ordinary unit and counit of the adjunction E M × M – : M - G -SSet τ ⇄ E M - G -SSet τ : forget are weak equivalences, so we concludeas in the proof of Theorem 2.1.41 that (2 . .
9) is a Quillen equivalence.By definition, (–) • : CMon( M - G -SSet τ ) → CMon( G - I -SSet τ ) preserves weakequivalences and fibrations, so (2 . .
10) is a Quillen adjunction. As also ev ω is ho-motopical, Theorem 1.4.41 implies by the same arguments as before that (2 . . M - G -SSet τ ) followsfrom left properness of the one on CMon( E M - G -SSet τ ) as E M × M – is leftQuillen and creates weak equivalences. (cid:3) The orthogonal spaces of Definition 1.5.1 admit a Day convolution product simi-larly to our models of G -global homotopy theory, i.e. there is an essentially uniquetopologically enriched functor– ⊠ – : L -Top × L -Top → L -Top that preserves cotensors and small colimits in each variable and satisfies L ( V, –) ⊠ L ( W, –) = L ( V ⊕ W, –) with the obvious functoriality in each variable. There is thenagain a unique way to make this into the tensor product of a symmetric monoidal
30 2. COHERENT COMMUTATIVITY structure on L -Top such that the structure isomorphisms on representables areinduced by the structure isomorphisms of the cartesian symmetric monoidal struc-ture on Vect R . Schwede introduced the term ultra-commutative monoid for thecommutative monoids in L -Top and he proved as [ Sch18 , Theorem 2.1.15-(i)]:
Theorem . There is a (unique) model structure on
CMon( L -Top ) inwhich a map is a weak equivalence or fibration if and only if it is so in the positiveglobal model structure on L -Top . This model structure is proper, topological, andcofibrantly generated. We now want to lift our comparison between I -SSet and L -Top to the levelof commutative monoids. For this we first observe that also I -Top admits a boxproduct extending the symmetric monoidal structure on I induced by disjoint union.As before one then makes | – | : I -SSet → I -Top and L × I – : I -Top → L -Top into strong symmetric monoidal functors. We conclude that their right adjointsare lax symmetric monoidal, so that we get induced adjunctions of categories ofcommutative monoids. We can now state our comparison result: Theorem . The adjunction (2.1.11) L × I | – | : CMon( I -SSet ) ⇄ CMon( L -Top ) : Sing ◦ forget is a Quillen adjunction, and the induced adjunction of associated quasi-categoriesis a (right Bousfield) localization with respect to the F in -global weak equivalences. Remark . Recently, Barrero Santamar´ıa [
Bar20 ] studied global E ∞ -operads in L -Top and proved that the homotopy theory of the corresponding alge-bras (with respect to the box product) is equivalent to the global homotopy theoryof ultra-commutative monoids (with respect to all compact Lie groups, and there-fore in particular with respect to finite groups), and hence by the results of thischapter also to all of the other models of ‘globally coherently commutative monoids’discussed above. Unfortunately, no non-trivial example of a global E ∞ -operad hasbeen established yet, although a candidate generalizing the little disks operad isdiscussed in [ Bar20 , Section 4].We will deduce Theorem 2.1.46 almost completely formally from the unstablecomparison provided in Section 1.5. However, there is a slight issue we have tosettle before: a cofibrant object M in CMon( G - I -SSet ) need not be cofibrant inthe positive G -global model structure on G - I -SSet . In fact, the initial objectof CMon( G - I -SSet ) is the terminal object of G - I -SSet , which is not positivelycofibrant. However we have: Lemma . Let M be cofibrant in CMon( G - I -SSet ) . Then M is cofibrantin the G -global model structure on G - I -SSet . Proof.
We will show that any cofibration X → Y in CMon( G - I -SSet ) with X cofibrant in the G -global model structure on G - I -SSet is also a G -global cofi-bration in G - I -SSet . Applying this to 0 → X then yields the lemma.To prove the claim we first observe that by the same calculation as in The-orem 2.1.33 the cofibrations of the G -global model structure satisfy the PushoutProduct Axiom and the Strong Commutative Monoid Axiom. While we cannotliterally apply [ Whi17 , Corollary 3.6] (because the acyclicity part of the StrongCommutative Monoid Axiom fails), the same argument works in our situation, asobserved for example in [
Sch18 , proof of Theorem 2.1.15-(ii)] for L -Top : .2. G -GLOBAL Γ-SPACES 131 Namely, it is clear by direct inspection that the forgetful functor sends generat-ing cofibrations of CMon( G - I -SSet ) to G -global cofibrations in G - I -SSet . If now f : X → Y is a pushout of a generating cofibration, then [ Whi17 , Proposition B.2](say, applied to the model structure on G - I -SSet in which all maps are cofibrationsbut only the isomorphisms are weak equivalences) shows that f can be written asa transfinite composition of maps of the form X ⊠ i (cid:3) nn / Σ n for generating cofibra-tions i n . By the Strong Commutative Monoid Axiom, each i (cid:3) nn / Σ n is a G -globalcofibration, and if X is cofibrant, so is X ⊠ i (cid:3) nn / Σ n by the pushout product axiom.Inductively we see that if X is cofibrant and X → Y is any relative cell complexin the generating cofibrations, then the underlying map in G - I -SSet is a G -globalcofibration. Finally, if f : X → Z is a general cofibration with cofibrant source,then Quillen’s Retract Argument shows that f is a retract of some relative cellcomplex g : X → Z (note that the sources agree!). As the underlying map of g is a G -global cofibration, so is the underlying map of f as desired. (cid:3) Proof of Theorem 2.1.46.
Analogously to Section 1.5 one shows that L × I | – | : I -SSet ⇄ L -Top : Sing ◦ forget is also a Quillen adjunction with respect tothe positive global model structures. We immediately conclude that (2 . .
11) is aQuillen adjunction.As weak equivalences are created in the underlying categories, we concludefrom Proposition 1.5.22 together with Proposition 1.5.16 that the right adjoint ishomotopical and inverts precisely the F in -global weak equivalences. It thereforeonly remains to show that the unit is a weak equivalence on any cofibrant object X ∈ CMon( I -SSet ). But by the previous lemma, X is cofibrant in the globalmodel structure on I -SSet , so the claim follows from Proposition 1.5.16 and The-orem 1.5.23. (cid:3) G -global Γ-spaces In this section we study a G -global version of Segal’s theory of Γ -spaces . Whilewe focus on them as models of ‘ G -globally coherently commutative monoids’ in thissection, we will discuss their relation to G -global spectra in Section 3.4. Let us briefly recall the classical non-equivariant and equivariant theory of Γ-spaces due to Segal [
Seg74 ] and Shi-makawa [
Shi89 ], respectively.
Definition . We write Γ for the category of finite pointed sets and basepoint preserving maps. For any n ≥
0, let n + := { , . . . , n } with base point 0.A Γ -space is a functor X : Γ → SSet such that X (0 + ) is terminal. We write Γ-SSet ∗ for the full subcategory of Fun(Γ , SSet ) spanned by the Γ-spaces.Segal [
Seg74 , Definition 1.2] originally considered contravariant functors froma category equivalent to the opposite of the above category Γ (into topologicalspaces), and he reserved the term ‘Γ-space’ for functors which are special in thesense of Definition 2.2.3 below.
Remark . The category
SSet ∗ of pointed simplicial sets has a zero object,so it admits a unique Set ∗ -enrichment; explicitly the base point of Hom( X, Y ) istaken to be the constant map. Similarly, there is a unique
Set ∗ -enrichment of Γ.As observed for example in [ MMO17 , Lemma 1.13] (for topological spaces),any
Set ∗ -enriched functor X : Γ → SSet ∗ satisfies X (0 + ) = ∗ , so its underlying
32 2. COHERENT COMMUTATIVITY ordinary functor Γ → SSet is a Γ-space. Conversely, every Γ-space X factorsuniquely through the category of SSet ∗ by taking the image of X (0 + ) as the basepoint of X ( S + ) for any S + ∈ Γ, and the induced functor Γ → SSet ∗ is Set ∗ -enriched [ MMO17 , Remark 1.15]. Altogether we see that we can equivalentlythink of
Γ-SSet ∗ as the category of Set ∗ -enriched functors Γ → SSet ∗ (with theisomormophism given by the forgetting base points and enrichment). Definition . Let S be a finite set and let s ∈ S . Then we write p s : S + → + for the map in Γ with p s ( s ) = 1 and p s ( t ) = 0 otherwise.Now let X be a Γ-space. The Segal map ρ : X ( S + ) → Y s ∈ S X (1 + )is the map given in the factor corresponding to s ∈ S by X ( p s ). We call X special if the Segal map is a weak homotopy equivalence for every finite set S .Intuitively, if X is a special Γ-space, then we want to think of X (1 + ) as its‘underlying space’ with the remaining structure encoding ‘(higher) additions.’ Ex-plicitly, consider for each n ≥ µ : n + → + sending every non-base pointto 1. Then we have a zig-zag X (1 + ) n X ( n + ) X (1 + ) ∼ ρ X ( µ ) which we think of as n -fold addition. In particular, for n = 2, passing to π yieldsa map π ( X (1 + )) ∼ = π ( X (1 + ) ) → π ( X (1 + )) and one can show that this equips π ( X (1 + )) with the structure of a commutative monoid. Example . If A is any simplicial abelian monoid, then we can definea special Γ-space HA via ( HA )( S + ) = L s ∈ S A ; writing elements of HA ( S + ) asformal sums P s ∈ S a s s , the structure map for f : S + → T + is given by P s ∈ S a s s P f ( s ) = ∗ a s f ( s ) = P t ∈ T (cid:0) P s ∈ f − ( t ) a s ) t . Example . Shimada and Shimakawa [
SS79 , Definition 2.1] showed howto functorially associate to a small symmetric monoidal category C a ‘special Γ-category,’ i.e. a functor Γ( C ) : Γ → Cat with Γ( C )(0 + ) = ∗ and such that theobvious analogues of the Segal maps Γ( C )( S + ) → Q s ∈ S Γ( C ) are equivalences ofcategories [ SS79 , Lemma 2.2]. In particular, applying the nerve levelwise yields aspecial Γ-space in the above sense.On the level of underlying categories, Γ( C ) recovers the original symmetricmonoidal category C , i.e. ev + ◦ Γ :
SymMonCat → Cat is (canonically) isomor-phic to the forgetful functor.
Remark . Let us call a morphism f : X → Y of Γ-spaces a level weakequivalence if each f ( S + ) : X ( S + ) → Y ( S + ) is a weak homotopy equivalence. Bous-field and Friedlander [ BF78 , Theorem 3.5] showed that the level weak equivalencesare part of a ‘Reedy type’ model structure on
Γ-SSet ∗ ; however, for our purposesthe usual projective model structure on Γ-SSet ∗ will be sufficient.For the rest of this subsection, let G be a finite group. Shimakawa [ Shi89 ]provided a G -equivariant generalization of Segal’s theory. Definition . A Γ - G -space is a functor X : Γ → G -SSet such that X (0 + )is terminal. We write Γ- G -SSet ∗ for the evident category of Γ- G -spaces. .2. G -GLOBAL Γ-SPACES 133 Shimakawa originally considered so-called ‘Γ G -spaces,’ suitably equivariant func-tors from the G -category Γ G of finite based G -sets and not necessarily G -equivariantmaps to G -SSet , but he showed in [ Shi91 , Theorem 1] that restricting alongΓ ֒ → Γ G provides an equivalence to the above category Γ- G -SSet ∗ . Remark . We can also make the quasi-inverse of this equivalence explicit:if X is any Γ- G -space, and S is a finite G -set, then X ( S + ) carries two commuting G -actions: the exterior action via the action of G on X and the interior action inducedby functoriality from the action on S , and we equip X ( S + ) with the diagonal ofthese two actions. This way, we can evaluate Γ- G -spaces more generally at finitepointed G -sets. A not necessarily G -equivariant map f : S + → T + of finite based G -sets then induces via the original functoriality in Γ a map X ( f ) : X ( S + ) → X ( T + )(not necessarily G -equivariant), which provides the desired extension to Γ G .We will only be interested in the case where f is indeed G -equivariant, in whichcase X ( f ) : X ( S + ) → X ( T + ) is obviously also G -equivariant.Shimakawa’s crucial insight was that while it is enough to specify Γ- G -spaceson trivial G -sets, the correct notion of specialness should still take general G -setsinto account: Definition . A Γ- G -space X is called special if the Segal map X ( S + ) → Q s ∈ S X (1 + ) is a G -equivariant weak equivalence for every finite G -set S . Here weequip the left hand side with the diagonal G -action as before, and the right handside carries the diagonal of the G -actions on X (1 + ) and the permutation action onthe factors via the G -action on S .We can also reformulate the above condition as saying that the Segal map X ( S + ) → Q S X (1 + ) should be a G G, Σ S -weak equivalence for every finite set S (note that this is indeed equivalent as any finite H -set S for H ⊂ G is an H -equivariant retract of the finite G -set G × H S ). Remark . The above strong version of specialness is necessary in orderfor special Γ- G -spaces to yield the correct notion of equivariant infinite loop spaces ,i.e. spaces that admit deloopings against all representation spheres; a particularlyclear explanation of this can be found in [ Blu06 , Section 3.5]. Moreover the ar-gument we give in 2.2.3.4 (for G -global Γ-spaces) shows how the above yields a Wirthm¨uller isomorphism for Γ- G -spaces, encoding an additional equivariant, or‘twisted,’ form of preadditivity. Example . If C is a small symmetric monoidal category with G -action,then we can equip the Γ-category S ( C ) from Example 2.2.5 with the induced G -action. However, the nerve of this is usually not special in the above sense.It was a crucial insight of Shimakawa [ Shi89 , discussion before Theorem A ′ ],recently extensively used by Merling [ Mer17 ], that we can solve this issue byreplacing the Γ- G -category S ( C ) with Fun( EG, –) ◦ S ( C ). Remark . Already when X is special in the na¨ıve sense, the zig-zag X (1 + ) × X (1 + ) X (2 + ) X (1 + ) ∼ ρ X ( µ ) equips π H ( X (1 + )) = π ( X (1 + ) H ) with an abelian monoid structure for all H ⊂ G .The special Γ- G -spaces are not invariant under the na¨ıve notion of levelwiseweak equivalences. Instead, one should require that f ( S + ) be a G -equivariant weakequivalence for every finite G -set S (and not only the trivial G -sets); put differently:
34 2. COHERENT COMMUTATIVITY
Definition . A map f : X → Y of Γ- G -spaces is called a G -equivariantlevel weak equivalence if f ( S + ) is a G G, Σ S -equivariant weak equivalence for everyfinite set S . Remark . As remarked by Ostermayr [
Ost16 , Theorem 4.7] withoutproof, the G -equivariant level weak equivalences are part of a ‘generalized projec-tive model structure’ on Γ- G -SSet . The fibrations in this model structure areprecisely those maps f such that f ( S + ) is a fibration in the G G, Σ S -model structureon ( G × Σ S )-SSet for all finite sets S . In addition, Ostermayr also constructs anequivariant version of the Bousfield-Friedlander model structure in [ Ost16 , Theo-rem 4.12]. G -global level model structures. In this subsection we will intro-duce various G -global analogues of Γ-spaces together with suitable level modelstructures. As the arguments for the existence of these model structures are rathersimilar, we will formalize them.For this we begin with the following ‘undirected’ version of [ Sch18 , PropositionC.23] (parts of which we recalled as Proposition 1.3.5 above):
Proposition . Let A be a small Set ∗ -enriched category and let C be alocally presentable category with -object (which is then Set ∗ -enriched in a uniqueway). Assume we are given for each a ∈ A a combinatorial model structure on Aut A ( a ) - C with sets of generating cofibrations I a and generating acyclic cofibrations J a , such that the following ‘consistency condition’ holds: ( ∗ ) for all b ∈ A , any relative { Hom A ( a, b ) ⊗ Aut A ( a ) j : a ∈ A, j ∈ J A } -cellcomplex is a weak equivalence in Aut B ( b ) - C .Then there exists a (necessarily unique) model structure on the category A - C of Set ∗ -enriched functors A → C in which a map f : X → Y is a weak equivalenceor fibration if and only if f ( a ) : X ( a ) → Y ( a ) is a weak equivalence or fibration,respectively, in Aut A ( a ) - C for each a ∈ A . We call this the generalized projectivemodel structure . It is combinatorial with generating cofibrations I A := { Hom( a, –) ⊗ Aut( a ) i : a ∈ A, i ∈ I a } and generating acyclic cofibrations. J A := { Hom( a, –) ⊗ Aut( a ) j : a ∈ A, j ∈ J a } . Moreover: (1)
If each of the model categories
Aut( a ) - C is simplicial, then so is A - C . (2) If each of the model categories
Aut( a ) - C is right proper, then so is A - C . (3) If filtered colimits are homotopical in each of the
Aut( a ) - C , then alsofiltered colimits in A - C are homotopical. In the above, we use for a pointed G - H -biset X the notation X ⊗ H Y forthe balanced tensor product , i.e. the quotient of the Set ∗ -tensoring X ⊗ Y by thediagonal H -action, with the induced G -action. In particular, in the world of pointedsimplicial sets X ⊗ H – can be identified with the usual balanced smash product X ∧ H –. Proof of Proposition 2.2.15.
For a ∈ A , let E a : A - C → Aut A ( a )- C de-note the evaluation functor. This admits a left adjoint G a , which by Kan’s pointwise .2. G -GLOBAL Γ-SPACES 135 formula (or the Yoneda Lemma) can be calculated as G a ( X )( b ) = Hom A ( a, b ) ⊗ Aut A ( a ) X with the obvious functoriality in each variable. Thus I A = [ a ∈ A G a ( I a ) and J A = [ a ∈ A G a ( J a ) . In order to construct the model structure in question and show that it is cofibrantlygenerated (hence combinatorial) with generating cofibrations I A and generatingacyclic cofibrations J A , it therefore suffices by Cran’s criterion, also cf. [ Ste16 ,Theorem A.1 and Remark A.2], to show:(1) The sets I A and J A permit the small object argument, and (2) Relative J A -cell complexes are weak equivalences (i.e. sent to weak equiv-alences in End A ( a )- C under E a for all a ∈ A ).The first condition is automatically satisfied as C and hence also A - C is locallypresentable. On the other hand, the second condition is an immediate consequenceof the consistency condition ( ∗ ).The additional properties (1)–(3) follow easily from the fact that all the relevantconstructions are defined levelwise. (cid:3) Instead of verifying the consistency condition by hand, we will employ thefollowing criterion:
Proposition . Let A be a small Set ∗ -enriched category and let C bea pointed locally presentable category. Assume we are given for each a ∈ A acofibrantly generated (hence combinatorial) model structure on Aut A ( a ) - C withsets of generating cofibrations I a and generating acyclic cofibrations J a , such that Aut A ( a ) - C is left proper and such that filtered colimits in it are homotopical. As-sume moreover: (1) For any a, b ∈ A and j ∈ J a , the map Hom( a, b ) ⊗ Aut( a ) j is a weakequivalence in Aut( b ) - C . (2) For any a, b ∈ A and i ∈ I a , any pushout along Hom( a, b ) ⊗ Aut( a ) i is ahomotopy pushout in Aut( b ) - C .Then the generalized projective model structure exists and is combinatorial with gen-erating cofibrations I A and generating acyclic cofibrations J A as above. Moreover,it is left proper and a square (2.2.1) W XY Z in A - C is a homotopy pushout if and only if it is a levelwise homotopy pushout,i.e. for every b ∈ A the induced square W ( b ) X ( b ) Y ( b ) Z ( b ) is a homotopy pushout in Aut( b ) - C . Finally, filtered colimits are homotopical in A - C , and the conclusions (1) and (2) of Proposition 2.2.15 still hold.
36 2. COHERENT COMMUTATIVITY
Proof.
We begin with the following observation:
Claim. If i is any I A -cofibration, then pushouts along i ( b ) are homotopypushouts in Aut( b )- C for any b ∈ A . In particular, pushouts along i in A - C are levelwise homotopy pushouts. Proof.
We fix b ∈ A and consider the class H b of all morphisms i in A - C such that pushouts along i ( b ) are homotopy pushouts. Using Proposition 1.1.13and that ev b is cocontinuous, we see that H b is closed under pushouts, transfinitecompositions, and retracts. On the other hand, I A ⊂ H b by assumption, so that H b contains all I A -cofibrations as desired. △ As each G a is cocontinuous, G a ( i ) is an I A -cofibration for any a ∈ A and anycofibration i in Aut( a )- C , so we in particular see that the second assumption holdsmore generally for all cofibrations i of Aut( a )- C .Let us now verify the assumption of Proposition 2.2.15, i.e. that relative J A -cellcomplexes are weak equivalences. As the latter are defined levelwise, this amountsto saying that for any b ∈ B any transfinite composition of pushouts of maps ofthe form Hom A ( a, b ) ⊗ Aut A ( a ) j ( a ∈ A, j ∈ J A ) is a weak equivalence. Indeed,any Hom A ( a, b ) ⊗ Aut A ( a ) j is a weak equivalence by the first assumption, and theabove strengthening of the second assumption then implies that also pushouts ofit are weak equivalences. The claim follows as filtered colimits were assumed to behomotopical.We are therefore allowed to apply Proposition 2.2.15, so it only remains to showthat A - C is left proper and that the homotopy pushouts are precisely the levelwisehomotopy pushouts. But the cofibrations of our model structure are precisely the I A -cofibrations, and as weak equivalences are defined levelwise, we immediatelyconclude from the claim that a pushout of a weak equivalence along a cofibrationin A - C is again a weak equivalence, i.e. A - C is left proper.Now assume we are given any square as in (2 . . W → X as acofibration W → H followed by a weak equivalence H → X . Then (2 . .
1) is bydefinition a homotopy pushout if and only if the induced map P := H ∐ W Y → Z is a weak equivalence, i.e. P ( b ) → Z ( b ) is a weak equivalence for every b ∈ B . Butthe latter fits into a commutative cube W ( b ) X ( b ) W ( b ) H ( b ) Y ( b ) Z ( b ) Y ( b ) P ( b )in Aut( b )- C . The front square is a homotopy pushout by the above claim and allthe front-to-back maps except possibly the lower right one are weak equivalencesfor obvious reasons. Thus the back square is a homotopy pushout if and only if alsothe lower right front-to-back map is a weak equivalence, which is precisely what wewanted to prove. (cid:3) Pointed G -global homotopy theory. The above model structures areinherently pointed, so in order to apply them to G -global homotopy theory, we .2. G -GLOBAL Γ-SPACES 137 need to developed suitable pointed versions of the models of Chapter 1. These arein fact formal consequences of the corresponding unbased results, and in order toavoid a long list of similar looking statements, we will be somewhat terse here.Let C be a category with terminal object ∗ . Then we write C ∗ := ∗ ↓ C forthe category of objects under ∗ and call it the category of pointed objects in C . If C has binary coproducts, then the forgetful functor C ∗ → C has a left adjoint (–) + sending an object X to X + = X ∐ ∗ with structure map the inclusion of the firstsummand, and similar on morphisms. Theorem . There is a unique model structure on E M - G -SSet ∗ inwhich a map is a weak equivalence, fibration, or cofibration if and only if it isso in the G -global model structure on E M - G -SSet . We call this the G -globalmodel structure again. It is proper, simplicial, and combinatorial with generatingcofibrations { ( E M × ϕ G × ∂ ∆ n ) + ֒ → ( E M × ϕ G × ∆ n ) + : H ⊂ M universal , ϕ : H → G, n ≥ } and generating acyclic cofibrations { ( E M× ϕ G × Λ nk ) + ֒ → ( E M× ϕ G × ∆ n ) + : H ⊂ M universal , ϕ : H → G, ≤ k ≤ n } . Moreover, filtered colimits in it are homotopical, and pushouts along injective cofi-brations are homotopy pushouts.
Proof.
The model structure exists by [
Hir03 , Theorem 7.6.5-(1)] applied tothe model structure from Corollary 1.2.30. Moreover, the model structure is cofi-brantly generated by the above sets according to [
Hir15 , Theorem 2.7], hence com-binatorial, and it is proper by [
Hir15 , Theorem 2.8-(3)]. Finally, filtered colimitsin E M - G -SSet ∗ are homotopical, as both weak equivalences as well as connectedcolimits are generated in E M - G -SSet . Similarly, Corollary 1.1.38 implies thatpushouts along injective cofibrations in E M - G -SSet ∗ are homotopical, so theyare homotopy pushouts as injective cofibrations are closed under pushouts.Finally, with respect to the evident simplicial enrichment E M - G -SSet ∗ istensored and cotensored over SSet with the cotensoring created in E M - G -SSet .It follows that the above model structure is simplicial. (cid:3) One gets analogous statements for all the other model structures established inChapter 1; instead of making them explicit, we will freely refer to the correspondingunbased statement whenever we actually need the based statement.If F : C ⇄ D : G is a Quillen adjunction such that F preserves the terminalobject, then we have an induced adjunction C ∗ ⇄ D ∗ , which is obviously a Quillenadjunction again. In particular, if α : H → G is any homomorphism, then wecan apply this to the various change of group adjunctions α ∗ ⊣ α ∗ discussed inChapter 1. Moreover, as weak equivalences are created in the corresponding modelsof unstable G -global homotopy theory, all results established above on whether α ∗ or α ∗ is fully homotopical, immediately transfer from the unpointed to the pointedsetting. We will therefore freely refer to the results from Chapter 1 when arguingabout the corresonding based adjunctions.While the functors α ! : H - C → G - C usually do not preserve the base point,the functor α ∗ : G - C ∗ → H - C ∗ still has a left adjoint in each case; as aboveone deduces that α ∗ is still right Quillen, so these are again Quillen adjunctions.The only results that do not immediately transfer via the above strategy are the
38 2. COHERENT COMMUTATIVITY criteria on when α ! is suitably homotopical. Let us therefore explicitly prove thecorresponding statements: Corollary . Let α : H → G be an injective homomorphism. Then α ! : E M - H -SSet ∗ → E M - G -SSet ∗ and α ! : E M - H -SSet τ ∗ → E M - G -SSet τ ∗ are homotopical. Proof.
It suffices to prove the first statement, for which we observe that α ! is left Quillen for the injective model structures by the above arguments appliedto Corollary 1.2.70. As all pointed E M - H -simplicial sets are injectively cofibrant,the claim follows from Ken Brown’s Lemma. (cid:3) Corollary . Let α : H → G be any homomorphism. Then α ! : E M - H -SSet ∗ → E M - G -SSet ∗ and α ! : E M - H -SSet τ ∗ → E M - G -SSet τ ∗ preserve weak equivalences between objects with free ker( α ) -action outside the basepoint. Proof.
It again suffices to prove the first statement, and the previous corollaryreduces to the case that α is surjective, i.e. α ! is given by dividing out the action of K := ker( α ). If we let L ⊂ M be any subgroup, then [ Hau17 , Lemma A.1] showsthat there is a natural isomorphism _ [ ψ : L → G ] Z ψ / (C G (im ψ ) ∩ K ) ∼ = ( Z/K ) ψ analogous to Proposition 1.1.45 whenever K acts freely on the Z outside the base-point. Analogously to the unbased case, C G (im ψ ) ∩ K acts freely on Z ψ outside thebase point, so Z ψ is cofibrant in the (non-equivariant) projective model structureon (C G (im ψ ) ∩ K ) -SSet ∗ . As the quotient functor (C G (im ψ ) ∩ K )-SSet ∗ → SSet ∗ is left Quillen, the claim follows. (cid:3) Remark . In fact the same argument as above more generally yields apointed version of Proposition 1.1.45.2.2.2.2.
Construction of G -global level model structures. With this at hand wecan now prove:
Theorem . There exists a unique model structure on Γ- E M - G -SSet ∗ in which a map f : X → Y is a weak equivalence or fibration if and only if f ( S + ) is a ( G × Σ S ) -global weak equivalence or fibration, respectively, for every finite set S . This model structure is proper, simplicial, and combinatorial with generatingcofibrations (cid:8)(cid:0) Γ( S + , –) ∧ ( E M × G ) + (cid:1) /H ∧ ( ∂ ∆ n ֒ → ∆ n ) + : H ∈ G U ,G , S finite H -set , n ≥ (cid:9) (where U denotes the collection of universal subgroups) and generating acyclic cofi-brations (cid:8)(cid:0) Γ( S + , –) ∧ ( E M× G ) + (cid:1) /H ∧ (Λ nk ֒ → ∆ n ) + : H ∈ G U ,G , S finite H -set , ≤ k ≤ n (cid:9) Moreover, filtered colimits in it are homotopical, and a square is a homotopy pushoutwith respect to it if and only if it so levelwise. In particular, pushouts along levelwiseinjections are homotopy pushouts. .2. G -GLOBAL Γ-SPACES 139 Proof.
One immediately proves by inspection that(2.2.2) Γ( A + , B + ) ∧ Σ A – : E M -( G × Σ A )-SSet ∗ → E M -( G × Σ B )-SSet ∗ sends the generating (acyclic) cofibrations of the usual ( G × Σ A )-global model struc-ture on the source to (acyclic) cofibrations in the injective ( G × Σ B )-global modelstructure on the target (where we for simplicity confuse Aut Γ ( S + ) with Σ S for anyfinite set S ). We may therefore apply Proposition 2.2.16, which proves all of theabove statements except for the description of the generating cofibrations.Instead, the aforementioned proposition shows that a set of generating cofibra-tions is given by the mapsΓ( S + , –) ∧ Σ S ( E M × G × Σ S × ∂ ∆ n ) + /H → Γ( S + , –) ∧ Σ S ( E M × G × Σ S × ∆ n ) + /H with H ∈ G U ,G × Σ S , n ≥ S an ordinary finite set. This is obviouslyconjugate to Γ( S + , –) ∧ Σ S ( E M × G × Σ S ) + ∧ ( ∂ ∆ n ֒ → ∆ n ) + ;on the other hand, Γ( S + , –) ∧ Σ S ( E M × G × Σ S ) + ∼ = Γ( S + , –) ∧ ( E M × G ) + via [ f, m, g, σ ] [ f σ, m, g ], which is right Σ S - and both left and right ( E M × G )-equivariant. As quotients and smash products preserve colimits, we concludeΓ( S + , –) ∧ Σ S ( E M × G × Σ S ) / Γ H, ( ϕ,ρ ) ∼ = (Γ( S + , –) ∧ ( E M × G ) + ) / Γ H, ( ϕ,ρ ) forany H ⊂ M and any homomorphisms ϕ : H → G and ρ : H → Σ S . Finally, if weview S as an Γ H,ϕ -set via ρ pr H , then the above is literally equal to (Γ( S + , –) ∧ ( EM × G ) + ) / Γ H,ϕ . This completes the verification of the given set of generatingcofibrations; the argument for the generating acyclic cofibrations is analogous. (cid:3)
Example . If | B | ≥
3, then (2 . .
2) is not left Quillen with respect tothe ordinary ( G × Σ B )-global model structure on the target. Namely, any cofibrantobject in the latter has free Σ B -action outside the basepoint, while Γ( A + , B + ) ∧ Σ A ( E M × Σ A ) + has non-trivial Σ B -fixed points: for example, if b ∈ B is arbitrary and β : A + → B + denotes the map sending every non-basepoint to b , then [ f, id ω , id A ]is fixed by the non-trivial subgroup of Σ bB ⊂ Σ B of the bijections fixing b .In particular, if f is a G -global cofibration, then f ( B + ) need not be a ( G × Σ B )-global cofibration for | B | ≥
3. However, one can at least show that it is a G -globalcofibration, cf. Proposition 2.2.25 below.We now want to prove a tame analogue of this: Theorem . There exists a (unique) model structure on Γ- E M - G -SSet τ ∗ in which a map f is a weak equivalence or fibration if and only if f ( S + ) is a positive( G × Σ S ) -global weak equivalence or fibration, respectively, for any finite set S .This model structure is combinatorial with generating cofibrations (cid:8)(cid:0) Γ( S + , –) ∧ ( E Inj( A, –) × G ) + (cid:1) /H ∧ ( ∂ ∆ n ֒ → ∆ n ) + : H finite group , S finite H -set , ∅ = A finite faithful H -set (cid:9) . Moreover, it is simplicial, proper, filtered colimits in it are homotopical, and asquare is a homotopy pushout if and only if it so levelwise. In particular, pushoutsalong levelwise injections are homotopy pushouts.
Here we use the positive model structure for technical reasons that will becomeapparent in later sections. To prove the theorem we will employ:
40 2. COHERENT COMMUTATIVITY
Lemma . Let i : Y → Z be an acyclic cofibration in the ( G × H ) -globalmodel structure on E M -( G × H )-SSet τ ∗ and let X be a left- H ′ -right- H -simplicialset. Then X ∧ H i is a ( G × H ′ ) -global weak equivalence. Proof.
Let K ⊂ M be any universal subgroup. By Lemma 1.4.39, i is a G K,G × H -cofibration, and it is obviously acyclic. Thus, it is enough to show that X ∧ H – sends acyclic cofibrations in the G K,G × H -model structure to acyclic cofibra-tions in the injective G K,G × H ′ -model structure. As before it suffices to prove thisfor the generating acyclic cofibrations, where this is trivial. (cid:3) Proof of Theorem 2.2.23.
By the same arguments as in Theorem 2.2.21 itsuffices that for any A + , B + ∈ Γ the functorΓ( A + , B + ) ∧ Σ A – : E M -( G × Σ A )-SSet τ ∗ → E M -( G × Σ B )-SSet τ ∗ sends positive ( G × Σ A )-global (acyclic) cofibrations to (acyclic) cofibrations in theinjective ( G × Σ B )-global model structure: indeed, it is clear that Γ( A + , B + ) ∧ Σ A i is an injective cofibration, and if i is acyclic, then the previous lemma implies thatΓ( A + , B + ) ∧ Σ A i is a ( G × Σ B )-global weak equivalence as desired. (cid:3) As before, if i is a cofibration in the above model structure, then i ( S + ) will notin general be a (positive) ( G × Σ S )-global cofibration for a finite set S with at leastthree elements. However we have: Proposition . Let f : X → Y be a cofibration in Γ- E M - G -SSet τ ∗ , andlet S be any finite set. Then f ( S + ) is a positive G -global cofibration. Proof.
It suffices to prove this for generating cofibrations, for which we arethen reduced by the same arguments as before to showing thatΓ( T + , S + ) ∧ ( G × E Inj(
A, ω ) + ) /H is cofibrant in the positive G -global model structure for any finite group A , anynon-empty faithful H -set A , and any homomorphism ϕ : H → G , where H actsfrom the right via its given action on A and its right action on G via ϕ .For this we first observe that the above is canonically isomorphic to (cid:0) E Inj(
A, ω ) + ∧ ( G + ∧ Γ( T + , S + )) (cid:1) /H. We now claim that the functor(2.2.3) ( H op × G )-SSet ∗ → E M - G -SSet τ ∗ X ( E Inj(
A, ω ) + ∧ X ) /H is left Quillen with respect to the G H op ,G -equivariant model structure on the sourceand the positive G -global model structure on the source. As the cofibrant objectson the source are precisely those simplicial sets with free G -action outside thebasepoint, this will then immediately imply the proposition.To prove the claim, we observe that (2 . .
3) factors up to isomorphism as thecomposition of the left Quillen functor ev ω and the functor ( H op × G )-SSet ∗ → G - I -SSet ∗ , X ( I ( A, –) + ∧ X ) /H which is left adjoint to X X ( A ) ϕ . The latter is by definition right Quillen withrespect to the G -global positive level model structure on the source, hence also withrespect to the G -global positive model structure. This completes the proof of theclaim and hence of the proposition. (cid:3) .2. G -GLOBAL Γ-SPACES 141 Corollary . Let X be cofibrant in the positive G -global model structureand let S be any finite set. Then: (1) X ( S + ) has no simplices of empty support except for the basepoint. (2) G acts freely on X ( S + ) outside the basepoint. Proof.
By the previous proposition, X ( S + ) is cofibrant in the positive G -global model structure. The two claims therefore follow from Lemma 1.4.39. (cid:3) Theorem . There exists a (unique) model structure on Γ- G - I -SSet ∗ inwhich a map f is a weak equivalence or fibration if and only if f ( S + ) is a ( G × Σ S ) -global weak equivalence or fibration, respectively, for any finite set S .This model structure is combinatorial with generating cofibrations (cid:8)(cid:0) Γ( S + , –) ∧ ( I ( A, –) × G ) + (cid:1) /H ∧ ( ∂ ∆ n ֒ → ∆ n ) + : H finite group , S finite H -set ,A finite faithful H -set (cid:9) . Moreover, it is simplicial, proper, filtered colimits in it are homotopical, and asquare is a homotopy pushout if and only if it so levelwise. In particular, pushoutsalong levelwise injections are homotopy pushouts.
Proof.
As before it suffices that Γ( A + , B + ) + ∧ Σ A – : ( G × Σ A )- I -SSet → ( G × Σ B )- I -SSet sends Σ A -global (acyclic) cofibrations to (acyclic) cofibrationsin the injective Σ B -global model structures for any A + , B + ∈ Γ.It i is a ( G × Σ A )-global cofibration, then it is clear that Γ( A + , B + ) ∧ Σ A i is an injective cofibration. Moreover, if i is acyclic, then i ( ω ) is an acyclic cofi-bration in the ( G × Σ A )-global model structure on E M -( G × Σ A )-SSet τ ∗ . As(Γ( A + , B + ) ∧ Σ A i )( ω ) is conjugate to Γ( A + , B + ) ∧ Σ A ( i ( ω )), the acyclicity parttherefore follows from Lemma 2.2.24. (cid:3) Remark . The analogues of the above theorem for the positive andrestricted G -global model structures hold and can be proven in the same way.Finally, we remark that similar arguments can be used to produce various G -globallevel model structures on M - G -SSet ∗ , M - G -SSet τ ∗ , and G - I -SSet ∗ ; as they willplay no role in the coming arguments, we leave the details to the interested reader.2.2.2.3. Comparison of level model structures.
Let us now lift some of the equiv-alences between the models of unstable G -global homotopy theory to Γ-spaces: Theorem . The functors (2.2.4) ev ω : Γ- G - I -SSet ∗ ⇄ Γ- E M - G -SSet ∗ : (–)[ ω • ] are homotopical and they descend to mutually inverse equivalences of associatedquasi-categories. Proof.
We first observe that each of the functors(–)[ ω • ] : E M -( G × Σ S )-SSet ∗ → ( G × Σ S )- I -SSet ∗ is homotopical by Proposition 1.3.28, so that also the right hand functor in (2 . . ω : ( G × Σ S )- I -SSet ∗ → E M -( G × Σ S )-SSet ∗ is homotopical by definition, hence so is the left hand functor in (2 . . ω • ] ◦ ev ω to the identity of I -SSet and shows that for any group H the induced
42 2. COHERENT COMMUTATIVITY transformation of endofunctors of H - I -SSet is a weak equivalence on all staticobjects. As both sides are homotopical, it then follows that it is actually a levelwiseweak equivalence and this then provides a levelwise weak equivalence exhibitingev ω as right inverse to (–)[ ω • ]. Analogously, we have constructed a natural zig-zag between ev ω ◦ (–)[ ω • ] and the identity of E M -SSet and we proved that forany H the induced zig-zag of endofunctors of E M - H -SSet is a levelwise weakequivalence. We conclude that the induced zig-zag also exhibits ev ω as left inverseof (–)[ ω • ] on the level of Γ-spaces. (cid:3) Remark . By the same arguments as in the proof of Theorem 2.2.27 thereis also a model structure on Γ- G - I -SSet ∗ based on the restricted model structures.It follows then immediately from the definitions that (–)[ ω • ] is right Quillen withrespect to this model structure, and hence part of a Quillen equivalence by theprevious theorem. However, as the restricted model structure is not relevant to thepresent article, we will not dwell on this.Similarly one shows: Theorem . The functors ev ω : Γ- G - I -SSet ∗ ⇄ Γ- E M - G -SSet τ ∗ : (–) • are homotopical and they descend to mutually inverse equivalences of associatedquasi-categories. (cid:3) By 2-out-of-3 we conclude from the above two theorems:
Corollary . The inclusion Γ- E M - G -SSet τ ∗ ֒ → Γ- E M - G -SSet ∗ in-duces an equivalence of associated quasi-categories. (cid:3) Connection to equivariant Γ -spaces. We now want to clarify the rela-tion between G -global Γ-spaces and the G -equivariant Γ-spaces discussed before.To this end, we introduce the following model structure which for finite G recov-ers [ Ost16 , Theorem 4.7] again:
Proposition . There is a unique model structure on Γ- G -SSet ∗ inwhich a map f is a weak equivalence or fibration if and only if f ( S + ) is a weak equiv-alence or fibration, respectively, in the G F in, Σ S -model structure on ( G × Σ S )-SSet .We call this the G -equivariant level model structure . It is simplicial, proper, andcombinatorial with generating cofibrations { (Γ( S + , –) ∧ G + ) /K ∧ ( ∂ ∆ n ֒ → ∆ n ) + : K ⊂ G finite, S finite K -set } . Finally, the G -equivariant level weak equivalences are stable under filtered colimits. Proof.
As before it suffices thatΓ( A + , B + ) ∧ Σ A – : ( G × Σ A )-SSet ∗ → ( G × Σ B )-SSet ∗ sends the standard generating (acyclic) cofibrations of the G F in, Σ A -model structureto (acyclic) cofibrations in the injective G F in, Σ A -model structure, which is clear bydirect inspection. (cid:3) The above condition is equivalent to demanding that f ( S + ) be an H -equivariantweak equivalence or fibration for any finite subgroup H ⊂ G and any finite H -set S . We caution the reader that unlike for finite G this is stronger than demandingthat f ( S + ) be a F in -weak equivalence or fibration for every finite G -set S : .2. G -GLOBAL Γ-SPACES 143 Example . Let f : X → Y be a level weak equivalence of Γ-spaces that isnot a Z /n -level weak equivalence (with respect to the trivial Z /n -action) for some n ≥
2. Then f is not a Q / Z -level weak equivalence either, as Z /n embeds into Q / Z . However, f ( S + ) is a Q / Z -weak equivalence for every finite Q / Z -set S , as Q / Z admits no non-trivial actions on finite sets.We will now discuss some homotopical properties of the passage from G -globalΓ-spaces to H -equivariant Γ-spaces along a group homomorphism ϕ : H → G , whichwill become crucial later in the proof of the G -Global Delooping Theorem. Whilesimilar results can be achieved for the other models introduced above, we willrestrict to the approach via I -simplicial sets. Definition . Let H be any group, let U H as in (1 . . ϕ : H → G be any homomorphism. Then the ϕ -underlying equivariant Γ -space of X ∈ Γ- G - I -SSet ∗ is ( ϕ ∗ X )( U H ). If f : X → Y is any map in Γ- G - I -SSet ∗ , then wedefine u ϕ ( f ) := ( ϕ ∗ f )( U H ).If H ⊂ G and ϕ is the inclusion, then we abbreviate u H := u ϕ . Lemma . Let f : X → Y be a map in Γ- G - I -SSet ∗ . Then f is a G -globalweak equivalence if and only if u ϕ ( f ) is an H -equivariant level weak equivalence forall finite groups H and all homomorphisms ϕ : H → G . Proof.
We may assume without loss of generality that H is a universal sub-group of M . Then both ω and U H are complete H -set universes, so u ϕ ( f ) is an H -equivariant weak equivalence if and only if ( ϕ ∗ f )( S + )( ω ) is an H -equivariantweak equivalence for all finite H -sets S ; here the H -action is via ϕ , ω , and S .If ρ : H → Σ S is the homomorphism classifying the H -action on S , then thisagrees with ( ϕ, ρ ) ∗ ( f ( S + ))( ω ) with f ( S + ) : X ( S + ) → Y ( S + ) viewed as a map in ( G × Σ S )- I -SSet ∗ . Thus, Corollary 1.2.74 implies that this is even an H -globalweak equivalence whenever f is a G -global level weak equivalence.Conversely, assume ( ϕ ∗ f )( S + )( ω ) = ( ϕ, ρ ) ∗ ( f ( S + ))( ω ) is an H -equivariantweak equivalence for all finite H as above, all ϕ : H → G , and all finite H -sets S . In particular, f ( S + )( ω ) ( ϕ,ρ ) is a weak equivalence; as every homomorphism ψ : H → G × Σ S is of the form ( ϕ, ρ ), we conclude that f ( S + ) is a ( G × Σ S )-globalweak equivalence as desired. (cid:3) We now want to give a simpler construction of u H under suitable fibrancyconditions. For this we will need: Corollary . There is a unique model structure on Γ- G - I -SSet ∗ inwhich a map f is • a cofibration if and only if each f ( S + ) is an injective cofibration, and • a weak equivalence if and only if each f ( S + ) is a ( G × Σ S ) -global weakequivalence.We call this the injective G -global level model structure . It is combinatorial, sim-plicial, proper, and filtered colimits in it are homotopical. Proof.
As pushouts along injective cofibrations preserve weak equivalences,this is an instance of Corollary A.2.11. (cid:3)
Warning . In general, fibrations in Γ- G - I -SSet ∗ need not be levelwiseinjective fibrations, i.e. ev T + : Γ- G - I -SSet ∗ → (Σ T × G )- I -SSet ∗ need not be
44 2. COHERENT COMMUTATIVITY right Quillen with respect to the injective G -global level and injective (Σ T × G )-global model structures. Equivalently, the left adjoint G T + need not be homotopical,as the following example for G = 1 shows: we let T be any finite set with | T | ≥ G T + were homotopical, then the same would be true for(2.2.5) Γ( T + , + ) ∧ Σ T – ∼ = ev + ◦ G T + : Σ T - I -SSet ∗ → I -SSet ∗ . However, the map f : T + → + with f ( t ) = 1 for all t ∈ T is Σ T -fixed, and so isthe map sending everything to the base point. We conclude that S with trivialΣ T -action is a Σ T -equivariant retract of Γ( T + , + ), so that (–) / Σ T ∼ = S ∧ Σ T – is aretract of (2 . . I ( T, –) + → ∗ + (where Σ T acts on the left hand side via its right action on T ) is a Σ T -global weak equivalence,and we claim that ( I ( T, –) / Σ T ) + → ∗ + is not a global weak equivalence. Indeed,it suffices that ( E Inj(
T, ω ) / Σ T ) → ∗ is not a global weak equivalence of E M -simplicial sets, but this isn’t even an underlying weak equivalence as E Inj(
T, ω ) / Σ T is a K (Σ T ,
1) and hence in particular not weakly contractible.However, we have:
Proposition . Let H ⊂ G and let S be a finite H -set. Then (2.2.6) ev S + : ( Γ- G - I -SSet ∗ ) injective G -global level → ( H - I -SSet ∗ ) injective H -global is right Quillen. Here X ( S + ) for X ∈ G -Γ- I -SSet ∗ is as usual equipped with thediagonal of the H -action on S and the restriction of the G -action on X . Proof.
We first observe that in the adjunction G + ∧ H – : Γ- H - I -SSet ∗ ⇄ Γ- G - I -SSet ∗ : res GH the left adjoint obviously preserves injective cofibrations and that it sends H -globallevel weak equivalences to G -global level weak equivalences by Lemma 1.3.54 (ap-plied to H × Σ S ֒ → G × Σ S for varying finite set S ). In particular, res GH is rightQuillen with respect to the injective level model structures, and we may thereforeassume without loss of generality that H = G .Let ρ : G → Σ S be the homomorphism classifying the G -action on S . Then(2 . .
6) factors as Γ- G - I -SSet ∗ ev S + −−−→ ( G × Σ S )- I -SSet ∗ (id ,ρ ) ∗ −−−−→ G - I -SSet ∗ , so a left adjoint is given by G S + ◦ (id , ρ ) ! . This obviously preserves injective cofi-brations, so it suffices to prove that it is homotopical, i.e. if f is a weak equivalenceof pointed G - I -simplicial sets, then(2.2.7) ( G S + (id , ρ ) ! f )( T + ) = Γ( S + , T + ) ∧ Σ S ( G × Σ S ) + ∧ G f is a ( G × Σ T )-global weak equivalence for every finite set T .Let us write Γ( S + , T + ) conj for Γ( S + , T + ) with G action via S and Σ T -action via T . Then an analogous calculation to the one from the proof of Theorem 2.2.21 showsthat (2 . .
7) is conjugate to Γ( S + , T + ) conj ∧ f . One easily checks that smashing withany pointed ( G × Σ T )-simplicial set sends G -global weak equivalences to ( G × Σ T )-global weak equivalences, which then concludes the proof. (cid:3) Corollary . Let X be fibrant in the injective G -global level model struc-ture, let H ⊂ G , and let i : A → B be an H -equivariant injection of H -sets. Then X (–)( A ) → X (–)( B ) is an H -equivariant level weak equivalence of Γ - H -spaces. .2. G -GLOBAL Γ-SPACES 145 Proof.
Let H ′ ⊂ H be finite and let S be a finite H ′ -set. We have to showthat X ( S + )( i ) : X ( S + )( A ) → X ( S + )( B ) is an H ′ -equivariant weak equivalence.But by the previous proposition X ( S + ) is an injectively fibrant H ′ - I -simplicial set,so the claim follows from Proposition 1.3.56. (cid:3) Corollary . Let X be fibrant in the injective G -global level model struc-ture on Γ- G - I -SSet ∗ and let H ⊂ G . Then ∅ → U H induces an H -equivariantlevel weak equivalence res GH X (–)( ∅ ) → X (–)( U H ) = u H ( X ) . (cid:3) Finally, let us lift the comparison between G - I -simplicial sets and G -simplicialsets established in Subsection 1.3.5 to the level of Γ-spaces: Proposition . The simplicial adjunctions (2.2.8) ev U G : Γ- G - I -SSet ∗ ⇄ Γ- G -SSet ∗ : R and (2.2.9) const : Γ- G -SSet ∗ ⇄ ( Γ- G - I -SSet ∗ ) injective : ev ∅ are Quillen adjunctions. Moreover, ev U G is homotopical, and u G = ev ∞U G ≃ R ev ∅ .Finally, the induced adjunctions ev ∞U G ⊣ R R and L const ⊣ R ev ∅ are a left andright Bousfield localization, respectively, with respect to the u G -weak equivalences. For the proof we will need:
Lemma . Let H be a finite group, let S be a finite H -set, and let i : X → Y be a cofibration in the G F in,H -model structure on ( G × H )-SSet ∗ . Then (Γ( S + , –) ∧ i ) /H is a cofibration in the G -equivariant level model structure on Γ- G -SSet ∗ . Proof.
As (Γ( S + , –) ∧ –) /H is cocontinuous and preserves tensors, it sufficesto prove that (Γ( S + , –) ∧ ( G × H ) + / Γ K,ϕ ) /H is cofibrant for any finite group K ⊂ G and any homomorphism ϕ : K → H . But indeed, this is isomorphic to (Γ( S + , –) ∧ G + ) /K where K acts on S via ϕ , and it acts on G in the obvious way. The claimfollows immediately from the description of the generating cofibrations given inProposition 2.2.33. (cid:3) Proof of Proposition 2.2.42. If f : X → Y is a G -global level weak equiv-alence and S is any finite set, then f ( S + ) is a ( G × Σ S )-global weak equivalenceby definition. In particular, if H ⊂ G is finite, then f ( S + ) is an H -global weakequivalence for any H -action on S . As U G contains a complete H -set universe(Lemma 1.3.58), we conclude from Lemma 1.3.11 that (ev U G f )( S + ) = ev U G ( f ( S + ))is an H -equivariant weak equivalence. This shows that ev U G is homotopical. Toprove that it is left Quillen, it is then enough by the previous lemma and the explicitdescription of the generating cofibrations, that E Inj( A, U G ) × G is G F in,H -cofibrantfor every finite faithful H -set A and H acting from the right on G via any homo-morphism ϕ . But H acts freely on the first factor, so every isotropy group belongsto G G,H ; on the other hand, G acts freely on the second factor, hence every isotropygroup belongs to G H,G , so it is in particular finite, as desired.For the proof that also (2 . .
9) is a Quillen adjunction, we let f : X → Y bea fibration or acyclic fibration in G - I -SSet ∗ ; we have to show that f ( S + )( ∅ ) isan fibration or acyclic fibration, respectively, in H -SSet ∗ for any H ⊂ G and anyfinite H -set S . But f ( S + ) is a fibration or acyclic fibration in the injective H -globalmodel structure by 2.2.39, so the claim follows from the proof of Corollary 1.3.57.
46 2. COHERENT COMMUTATIVITY
Next, we observe that the inclusion ∅ ֒ → U G induces a natural transformation ι : ev ∅ ⇒ ev U G , which is a G -equivariant weak equivalence on injectively fibrant G -global Γ-spaces by Corollary 2.2.41. As ev U G is homotopical, this yields the desiredequivalence (ev U G ) ∞ ≃ R ev ∅ .It only remains to show that the derived unit X → R ev ∅ (const X ) is a G -equivariant weak equivalence for every (cofibrant) equivariant Γ-space X . For thiswe fix an injectively fibrant replacement κ : const X → Y and consider the commu-tative diagram X ev ∅ const X ev U G const X ev ∅ Y ev U G Y η ∼ = ι ∼ =ev ∅ κ ev U G κ ∼∼ ι The composition X → ev ∅ Y represents the derived unit. But the top horizontalarrows are isomorphisms by direct inspection, the lower horizontal arrow is a G -equivariant weak equivalence by the above, and the right hand vertical map isa G -equivariant weak equivalence as ev U G is homotopical. The claim follows by2-out-of-3. (cid:3) Warning . The functor const : Γ- G -SSet ∗ → Γ- G - I -SSet ∗ is not ho-motopical with respect to the G -global level weak equivalences at the target, butit is homotopical with respect to the u G -weak equivalences.2.2.2.5. Functoriality.
The results on the change-of-group adjunctions for themodels of (pointed) unstable G -global homotopy theory discussed in Chapter 1easily transfer to statements about the corresponding adjunctions on the level ofΓ-spaces. Instead of making all of these explicit, we will only collect the resultshere that we will need below: Corollary . Let α : H → G be any group homomorphism. Then theadjunction α ! : Γ- E M - H -SSet τ ∗ ⇄ Γ- E M - G -SSet τ ∗ : α ∗ is a Quillen adjunction with fully homotopical right adjoint. Moreover, if α isinjective, then also the left adjoint is homotopical. Proof.
As weak equivalences and fibrations are defined levelwise, this followsby applying Lemma 1.4.44 to α × Σ S for varying finite set S . (cid:3) Similarly one deduces from Lemma 1.4.46:
Corollary . Let α : H → G be an injective homomorphism with ( G :im α ) < ∞ . Then the adjunction α ∗ : Γ- E M - G -SSet τ ∗ ⇄ Γ- E M - H -SSet τ ∗ : α ∗ is a Quillen adjunction in which both functors are homotopical. (cid:3) G -global Γ-spaces. Just as in the classical equivariant ornon-equivariant setting, we want to think of the maps X ( S + ) → X (1 + ) induced bythe unique maps S
7→ { } as ‘generalized multiplications.’ In order for this intuitionto apply, we need to Bousfield localize at a suitable class of G -global Γ-spaces: .2. G -GLOBAL Γ-SPACES 147 Definition . We call X ∈ Γ- E M - G -SSet ∗ special if the followingholds: for all finite sets S the Segal map X ( S + ) ρ := X ( p s ) s ∈ S −−−−−−−−→ Y s ∈ S X (1 + )is a ( G × Σ S )-global weak equivalence, where Σ S acts on both sides via its tautolog-ical action on S , and analogously for X ∈ Γ- E M - G -SSet τ ∗ or X ∈ Γ- G - I -SSet ∗ .We denote the corresponding full subcategories by the superscript ‘special.’One easily proves by direct inspection similarly to Lemma 2.2.36: Lemma . A G -global Γ -space X ∈ Γ- G - I -SSet ∗ is special if and onlyif u ϕ X is special in the sense of Definition 2.2.9 for all finite groups H and allhomomorphisms ϕ : H → G . (cid:3) Moreover, we can conclude from the equivalences discussed above:
Corollary . All the (homotopical) functors in the diagram Γ- E M - G -SSet τ, special ∗ Γ- E M - G -SSet special Γ- G - I -SSet special ∗ (–)[ ω • ]ev ω induce equivalencess of associated quasi-categories. Moreover, the resulting diagramcommutes up to preferred equivalence. Proof.
As specialness is obviously invariant under G -global level equivalences,Proposition A.1.15 implies that for all of the above models the inclusion of the fullsubcategory of special G -global Γ-spaces descends to a fully faithful functor ofassociated quasi-categories.Next we observe that all of the functors are defined levelwise. The correspond-ing functors preserve and reflect ( G × Σ S )-global weak equivalences (as they arehomotopical and induce equivalences by the results of Chapter 1), and they pre-serve finite products by direct inspection. Thus, we conclude that they preserveand reflect specialness.Thus, the comparisons of G -global Γ-spaces from the previous section impliesthat all of the above functors descend to equivalences of quasi-localizations. It onlyremains to show the commutativity up to equivalence, which now follows easilyfrom Theorem 2.2.29. (cid:3) A model categorical manifestation.
For all of our above model cate-gories of G -global Γ-spaces, one can obtain a model of special G -global Γ-spacesvia Bousfield localization. We will make this explicit for Γ- E M - G -SSet τ ∗ : Definition . A morphism f : X → Y in Γ- E M - G -SSet τ ∗ is called a special G -global weak equivalence if the induced map f ∗ : [ Y, T ] → [ X, T ] is bijectivefor all special G -global Γ-spaces T . Here [ , ] denotes the hom sets in the homotopycategory of Γ- E M - G -SSet τ ∗ with respect to the G -global level weak equivalences. Theorem . There exists a unique model structure on Γ- E M - G -SSet τ ∗ with the same cofibrations as the positive G -global level model structure, and weakequivalences the G -global special weak equivalences. Its fibrant objects are precisely
48 2. COHERENT COMMUTATIVITY the positively level fibrant special G -global Γ -spaces, and we call this the special G -global model structure .It is left proper, simplicial, and filtered colimits in it are homotopical. More-over, it is combinatorial and there exists a set J of generating acyclic cofibrationsconsisting only of maps between cofibrant objects. Proof.
For the existence of the model structure, and the proof that it issimplicial, left proper, and combinatorial, it suffices by Theorem A.2.4 that thereexists a set T of maps between cofibrant objects such that the special G -globalΓ-spaces are precisely the T -local objects Z , i.e. those such that the induced mapmaps( f, Z ′ ) : maps( Y, Z ′ ) → maps( X, Z ′ ) is a weak homotopy equivalence of simpli-cial sets for all f : X → Y in T and some (hence any) choice of fibrant replacement Z ′ of Z in the positive G -global model structure.For this we let H be a universal group of M , S a finite set, and ϕ : H → G , ρ : H → Σ S be homorphisms. We fix a free H -orbit F ⊂ ω . Then (cid:0) Γ( S + , –) ∧ ( E Inj(
F, ω ) × G × Σ S ) + (cid:1) /H (where H acts on G from the right via ϕ and on Σ S via ρ , and on F in the tautological way) corepresents the functor X X ( S + ) ( ϕ,ρ )[ F ] in the simplicially enriched sense. Here H acts via ϕ , ρ and the restriction of the E M -action to H (which preserves simplices supported on the H -set F ).Similarly, ( S + ∧ Γ(1 + , –) ∧ ( E Inj(
F, ω ) × G ) + ) /H corepresents the functor X (cid:0) Q s ∈ S X (1 + ) (cid:1) ( ϕ,ρ )[ F ] . By the Yoneda Lemma we therefore get a map λ H,S,ϕ,ρ : (cid:0) S + ∧ Γ(1 + , –) ∧ ( E Inj(
F, ω ) × G ) + (cid:1) /H → (cid:0) Γ( S + , –) ∧ ( E Inj(
F, ω ) × G ) + (cid:1) /H such that for any Z the restriction maps( λ H,S,ϕ,ρ , Z ) is conjugate to( p S ) ( ϕ,ρ )[ F ] : Z ( S + ) ( ϕ,ρ )[ F ] → Y s ∈ S Z (1 + ) ! ( ϕ,ρ )[ F ] ;explicitly, λ H,S,ϕ,ρ is induced by the map(2.2.10) S + ∧ Γ(1 + , T + ) → Γ( S + , T + ) , [ s, f ] f ◦ p s Now assume Z fibrant in the G -global positive level model structure. Then Z ( S + )is fibrant in the positive ( G × Σ S )-global model structure, so the inclusion induces aweak equivalence Z ( S + ) ( ϕ,ρ )[ F ] ֒ → Z ( S + ) ( ϕ,ρ ) by Remark 1.4.38. On the other hand, Q s ∈ S Z (1 + ) is fibrant in the ( G × Σ S )-global model structure by Corollary 1.4.47;we conclude by the same argument as before that the inclusion induces a weakequivalence (cid:0)Q s ∈ S Z (1 + ) [ F ] (cid:1) ( ϕ,ρ ) → (cid:0)Q s ∈ S Z (1 + ) (cid:1) ( ϕ,ρ ) . Altogether we see that p ( ϕ,ρ ) S is a weak equivalence if and only if maps( λ H,S,ϕ,ρ , Z ) is a weak equivalence.In particular, p S is a G -global weak equivalence if and only if maps( λ H,S,ϕ,ρ , Z ) isa weak equivalence for all ϕ and ρ as above.Now we define T to be set of all λ H,S,ϕ,ρ for all universal H ⊂ M , all S = { , . . . , n } for n ≥
0, and all homomorphisms ϕ : H → G , ρ : H → Σ S . The abovethen shows that a positively level fibrant Z is special if and only if maps( f, Z ) isa weak equivalence for all f ∈ T . On the other hand, the targets of the mapsin S are obviously cofibrant, and for the sources it suffices to observe that thefunctor corepresented by them sends acyclic G -global level cofibrations to acyclicKan fibrations by Corollary 1.4.47. This completes the proof of the existence of themodel structure. .2. G -GLOBAL Γ-SPACES 149 Lemma A.2.6 immediately implies that filtered colimits in the resulting modelstructure are homotopical. Finally, if I is the usual set of generating cofibrationsof the positive G -global level model structure, then I is also a set of generatingcofibrations for the new model structure. As I obviously consists of maps betweencofibrant objects, [ Bar10 , Corollary 2.7] implies that also the set J of generating acyclic cofibrations can be chosen to consist of maps between cofibrant objects. (cid:3) Lemma . Let f : X → Y be a map in Γ- E M - G -SSet τ, special ∗ . Then thefollowing are equivalent: (1) f is a G -global special weak equivalence. (2) f is a G -global level weak equivalence. (3) f (1 + ) is a G -global weak equivalence. Proof.
It is clear that (2) ⇒ (1) and (2) ⇒ (3). The implication (1) ⇒ (2)is a general fact about Bousfield localizations, using that the special objects areprecisely the local ones. Finally, if S is any finite set, then f ( S + ) agrees with Q s ∈ S f up to conjugation by ( G × Σ S )-global weak equivalences. The implication(3) ⇒ (2) thus follows from Corollary 1.4.47. (cid:3) Lemma . Let f : X → Y be an injective cofibration in Γ- E M - H -SSet τ ∗ .Then any pushout along f is a homotopy pushout in the G -global special modelstructure. Proof.
The G -global level weak equivalences are stable under pushouts alonginjective cofibrations by Theorem 2.2.23. The claim therefore follows as in the proofof Theorem 1.2.41. (cid:3) Proposition . Let α : H → G be any group homomorphism. Then thesimplicial adjunction α ! : ( Γ- E M - H -SSet τ ∗ ) special ⇄ ( Γ- E M - G -SSet τ ∗ ) special : α ∗ is a Quillen adjunction. If α is injective, then α ! is fully homotopical. Proof.
For the corresponding level model structures, this was shown as Corol-lary 2.2.45. To prove the first statement, it is then enough to show that α ∗ preservesspecialness, which is immediate from Lemma 1.4.44.The second statement then follows from Corollary 2.2.45 as any special G -global weak equivalence factors as a special G -global acyclic cofibration followed bya G -global level weak equivalence. (cid:3) G -global vs. G -equivariant specialness. We now want to use the com-parison between G -global and G -equivariant Γ-spaces from Proposition 2.2.42 toprove: Theorem . The homotopical functor ev U G : Γ- G - I -SSet ∗ → Γ- G -SSet ∗ induces a quasi-localization ( Γ- G - I -SSet special ∗ ) ∞ → ( Γ- G -SSet special ∗ ) ∞ . The proof of this is slightly more involved because being G -globally specialis a much stronger condition than being G -equivariantly special; in particular, itturns out that const: Γ- G -SSet ∗ → Γ- G - I -SSet ∗ does not preserve specialness.Instead, we will have to consider the right adjoint R of ev ∞U G , but proving that thishas the desired properties requires some preparations.We begin with the following analogue of Theorem 2.2.51:
50 2. COHERENT COMMUTATIVITY
Proposition . There is a unique model structure on Γ- G -SSet ∗ withthe same cofibrations as the G -equivariant level model structure, and whose fibrantobjects are precisely the G -equivariant level fibrant special Γ - G -spaces. This modelstructure is combinatorial, simplicial, left proper, and filtered colimits in it arehomotopical. Proof.
This is proven in precisely the same way as Theorem 2.2.51 by local-izing with respect to the maps(2.2.11) ( S + ∧ Γ(1 + , –) ∧ G + ) /H → (Γ( S + , –) ∧ G + ) /H induced by (2 . .
10) for every finite H ⊂ G and every finite H -set S ; note that thesources are indeed cofibrant as they corepresent X maps ∗ ( S + , X (1 + )) H whichis isomorphic to X Q ri =1 X (1 + ) K i when S = ` ni =1 H/K i . (cid:3) We call the weak equivalences of the above model structure the G -equivariantspecial weak equivalences . They can again be detected by mapping into special Γ- G -spaces; in particular, each of the maps (2 . .
11) is a G -equivariant special weakequivalence. Proposition . There is a unique model structure on Γ- G - I -SSet ∗ withthe same cofibrations as the G -global level model structure, and whose fibrant ob-jects are precisely the G -globally level fibrant special G -global Γ -spaces. This modelstructure is combinatorial, simplicial, left proper, and filtered colimits in it are ho-motopical. Proof.
This follows similarly by localizing with respect to the maps λ H,S,ϕ,ρ : (cid:0) S + ∧ Γ(1 + , –) ∧ ( I ( H, –) × G ) + (cid:1) /H → (cid:0) Γ( S + , –) ∧ ( I ( H, –) × G ) + (cid:1) /H analogous to the above. (cid:3) Lemma . Let K be any group, and let X ∈ ( G × K ) -SSet ∗ be cofibrantin the G F in,K -equivariant model structure. Then the map ( S + ∧ Γ(1 + , –) ∧ X ) /K → (Γ( S + , –) ∧ X ) /K induced by (2 . . is a G -equivariant special weak equivalencebetween cofibrant objects for every finite K -set S . Proof.
We first note that the target is indeed cofibrant by Lemma 2.2.43 andso is the source by the following observation:
Claim.
The functor ( S + ∧ Γ(1 + , –) ∧ –) /K : ( G × K )-SSet ∗ → Γ- G -SSet ∗ sends G F in,K -cofibrations to cofibrations. Proof.
As in Lemma 2.2.43 it suffices to show that ( S + ∧ Γ(1 + , –) ∧ ( G × K ) + / Γ H,ϕ ) /K is cofibrant for any finite H ⊂ G and any homomorphism ϕ : H → K . But this is isomorphic to (( ϕ ∗ S ) + ∧ Γ(1 + , –) ∧ G ) /H , so the claim follows fromthe proof of Proposition 2.2.56. △ Fix K and S ; we will verify the conditions of Corollary 1.1.8 for the naturaltransformation τ : ( S + ∧ Γ(1 + , –) ∧ –) /K ⇒ (Γ( S + , –) ∧ –) /K induced by (2 . . X = ( G × K ) / Γ H,ϕ for some finite subgroup H ⊂ G and some homomorphism ϕ : H → K , then τ X is easily seen to be conjugate to the map ( ϕ ∗ S + ∧ Γ(1 + , –) ∧ –) /H → (Γ( ϕ ∗ S + , –) ∧ –) /H induced by (2 . . G -equivariant specialweak equivalence by the above. Moreover, if Y is any simplicial set, then τ X × Y is conjugate to τ X × Y , so τ is a G -equivariant special weak equivalence for ( G × K ) / Γ H,ϕ × Y as the G -global special model structure is simplicial. In particular, τ .2. G -GLOBAL Γ-SPACES 151 is a G -global special weak equivalences on the sources and targets of the generatingcofibrations. The claim now follows as ( S + ∧ Γ(1 + , –) ∧ –) /K and (Γ( S + , –) ∧ –) /K each preserve cofibrations as well as small colimits. (cid:3) Corollary . The simplicial adjunction (2.2.12) ev U G : ( Γ- G - I -SSet ∗ ) G -global special ⇄ ( Γ- G -SSet ∗ ) G -equiv. special : R is a Quillen adjunction with fully homotopical left adjoint. Proof.
Let us first show that this is a Quillen adjunction. Since we alreadyknow this for the respective level model structures (Proposition 2.2.42), it sufficesto show that R sends special Γ- G -spaces to special G -global Γ-spaces.For this we let H be any finite group and F be any finite free H -set. Then wehave seen in the proof of Proposition 2.2.42 that ( E Inj( H, U G ) × G ) + is cofibrant inthe G F in,H -model structure, so the map ( S + ∧ Γ(1 + , –) ∧ ( E Inj( H, U G ) × G ) + ) /H → (Γ( S + , –) ∧ ( E Inj( H, U G ) × G ) + ) /H induced by (2 . .
10) is a G -equivariant specialweak equivalence of cofibrant objects according to the previous lemma. The claimnow follows from an easy adjointness argument.We conclude in particular that ev U G sends G -global special acyclic cofibrationsto G -equivariant special weak equivalences. As any G -global special weak equiva-lence factors as an acyclic cofibration followed by a G -global level weak equivalence,we conclude together with Proposition 2.2.42 that ev U G is homotopical. (cid:3) Proof of Theorem 2.2.55.
It is easy to check that ev U G sends special G -global Γ-spaces to special Γ- G -spaces. In the commutative diagram( Γ- G - I -SSet ∗ ) special G -glob. level ( Γ- G - I -SSet ∗ ) G -glob. special ( Γ- G -SSet ∗ ) special G -equiv. level ( Γ- G -SSet ∗ ) G -equiv. levelev U G ev U G of homotopical functors, the horizontal maps induce equivalences of associatedquasi-categories; it therefore suffices to show that (2 . .
12) induces a Bousfield lo-calization of associated quasi-categories.Since ev U G is homotopical, this amounts to demanding that the ordinary counit ǫ : ev U G RX → X be a G -equivariant special weak equivalence for every X ∈ Γ- G -SSet ∗ fibrant in the G -equivariant special model structure. But any such X is in particular fibrant in the G -equivariant level model structure, so ǫ is even a G -equivariant level weak equivalence by Proposition 2.2.42. (cid:3) Preadditivity.
We want to think of special G -global Γ-spaces as homoto-py coherent versions of commutative monoids, so at the very least their homotopycategory should be preadditive. In this subsection we will prove the following ‘un-derived’ version of this: Theorem . The canonical map ι : X ∨ Y → X × Y is a G -global specialweak equivalence for all X, Y ∈ Γ- E M - G -SSet τ ∗ . The proof we will give below is a spiced-up version of the usual argument thatfinite coproducts and products in a category of commutative monoids agree. Thisrequires some preparations.
52 2. COHERENT COMMUTATIVITY
Construction . Let
A, B, C ∈ Γ, and let f : B → C be any morphism.Then we have an induced map A ∧ f : A ∧ B → A ∧ C . If C = 1 + , then we will byslight abuse of notation also denote the composition A ∧ B → A ∧ C ∼ = A with thecanonical isomorphism A ∧ C ∼ = A, [ a, a by A ∧ f , and similarly for B = 1 + .Now let T ∈ Γ- E M - G -SSet τ ∗ . Applying the above to the map µ : 2 + → + , µ (1) = µ (2) = 1 induces T ( µ ∧ –) : T (2 + ∧ –) → T , and this is clearly naturalin T . On the other hand, the assignment T T (2 + ∧ –) obviously preserves G -global level weak equivalences, so it descends to a functor on the homotopy categoryHo( Γ- E M - G -SSet τ ∗ ) strict with respect to these. It follows formally that the map µ : T (2 + ∧ –) → T is also natural with respect to maps in the homotopy category.Similarly the Segal maps assemble into a natural morphism ρ := ( T ( p ∧ –) , T ( p ∧ –)) : T (2 + ∧ –) → T × T ;as the product preserves G -global level weak equivalences, this again descends to anatural transformation on the strict homotopy category.The following lemma follows easily from the definitions and we omit its proof. Lemma . Let T ∈ Γ- E M - G -SSet τ ∗ be special. Then ρ : T (2 + ∧ –) → T × T is a G -global level weak equivalence. (cid:3) If T is special, then we write m T for the map T × T → T in the strict homotopycategory corresponding to the zig-zag T × T T T. ∼ ρ T ( µ ∧ –) Proof of Theorem 2.2.60.
Let us fix a special T ∈ Γ- E M - G -SSet τ ∗ ; wehave to show that ι ∗ : [ X × Y, T ] → [ X ∨ Y, T ] is bijective, where [ , ] denotes homsets in the strict homotopy category.As ∨ is homotopical in G -global level weak equivalences, X ∨ Y is also a coprod-uct in the strict homotopy category, so that ( i ∗ X , i ∗ Y ) : [ X ∨ Y, T ] → [ X, T ] × [ Y, T ]is bijective. It therefore suffices to show that the composition α : [ X × Y, T ] → [ X, T ] × [ Y, T ] is bijective; plugging in the definition, we see that this is given by( j ∗ X , j ∗ Y ), with j X : X → X × Y , j Y : Y → X × Y the inclusions.To prove the claim, we define β : [ X, T ] × [ Y, T ] → [ X × Y, T ] as follows: if f ∈ [ X, T ] , g ∈ [ Y, T ] are arbitrary, then β ( f, g ) := m T ◦ ( f × g ); note that thisindeed makes sense because × descends to the homotopy category by the above.We claim that β is a two-sided inverse of α . Indeed, if f ∈ [ X, T ] , g ∈ [ Y, T ] arearbitrary, then αβ ( f, g ) = α ( m T ◦ ( f × g )) = ( m T ◦ ( f × g ) ◦ j X , m T ◦ ( f × g ) ◦ j Y ). Wewill prove that m T ◦ ( f × g ) ◦ j X = f , the argument for the second component beingsimilar. Indeed, we have ( f × g ) j X = j f as both sides agree after postcomposingwith any of the two projections T × T → T (here we again use that T × T is aproduct in the homotopy category). On the other hand, let us consider the diagram(2.2.13) TT T (2 + ∧ –) T × T = T ( i ∧ –) j T ( µ ∧ –) ρ .2. G -GLOBAL Γ-SPACES 153 where i : 1 + → + is defined by i (1) = 1. Then the top triangle commutesbecause µi = id + , and we claim that also the lower triangle commutes. Indeed,after postcomposing with the projection to the first factor both paths through thediagram are the identity (as p i = id + ). On the other hand, pr j is constantat the basepoint, and so is pr ρT ( i ∧ –) = T ( p ∧ –) ◦ T ( i ∧ –) since it factorsthrough T (0 + ) = ∗ . From commutativity of (2 . .
13) we can now conclude m T j = µT ( i ∧ –) = id, hence m T ◦ ( f × g ) ◦ j X = f as desired.Finally, let F ∈ [ X × Y, T ] be arbitrary. Then βα ( F ) = β ( F j X , F j Y ) = m T ◦ (cid:0) ( F j X ) × ( F j Y ) (cid:1) . We now consider the diagram(2.2.14) ( X × Y ) × ( X × Y ) T × TX × Y X (2 + ∧ –) × Y (2 + ∧ –) T (2 + ∧ –) X × Y T F × F = j X × j Y X ( i ∧ –) × Y ( i ∧ –) ρX ( µ ∧ –) × Y ( µ ∧ –) F (2 + ∧ –) ρT ( µ ∧ –) F in the strict homotopy category. The right hand portion commutes by the naturalityestablished in Construction 2.2.61, and as above one shows that the triangles onthe left already commute on the pointset level.Using the commutativity of (2 . .
14) we then compute βα ( F ) = m T ◦ (cid:0) ( F j X ) × ( F j Y ) (cid:1) = m T ◦ ( F × F )( j X × j Y )= m T ◦ ρ ◦ F (2 + ∧ –) ◦ (cid:0) X ( i ∧ –) × Y ( i ∧ –) (cid:1) = T ( µ ∧ –) ◦ F (2 + ∧ –) ◦ (cid:0) X ( i ∧ –) × Y ( i ∧ –) (cid:1) = F, which completes the proof of the theorem. (cid:3) Corollary . Finite coproducts and finite products in Γ- E M - G -SSet τ ∗ preserve G -global special weak equivalences. Proof.
It suffices to treat the case of binary coproducts and products, forwhich we let f : X → X ′′ and g : Y → Y ′′ be any weak equivalences. Then wecan factor f as an acyclic cofibration i : X → X ′ followed by an (automaticallyacyclic) fibration p : X ′ → X ′′ and similarly g = qj with an acyclic fibration q andan acyclic cofibration j . In the commutative diagram X ∨ Y X ′ ∨ Y ′ X ′′ ∨ Y ′′ X × Y X ′ × Y ′ X ′′ × Y ′′ i ∨ jι ι p ∨ q ιi × j p × q the vertical arrows are weak equivalences by the previous theorem, and so are thetop left and bottom right arrows as acyclic cofibrations in any model category arestable under (all) coproducts while acyclic fibrations are stable under products.The claim follows by 2-out-of-3. (cid:3) Warning . We can define specialness for elements of Γ- E M - G -SSet in the same way as above, which leads to a notion of G -global special weak equiv-alences on Γ- E M - G -SSet . These are however not stable under finite products:
54 2. COHERENT COMMUTATIVITY for example the map f : (Γ(1 + , –) ∐ Γ(1 + , –)) × E Inj( ∗ , ω ) × G → Γ(2 + , –) × E Inj( ∗ , ω ) × G induced by restricting along p , p is a G -global special weak equivalence, becausefor any level fibrant T the induced map [ f, T ] is conjugate to the map π T (2 + )( ω ) → π T (1 + ) × π T (1 + )( ω ) induced by the Segal map. However, Γ(1 + , –) × f is conjugateto a map (Γ(2 + , –) ∐ Γ(2 + , –)) × E Inj( ∗ , ω ) × G → Γ(3 + , –) × E Inj( ∗ , ω ) × G by theuniversal property of coproducts in Γ, hence not a G -global special weak equivalenceby a similar calculation as before.Thus, the fact that also non-special G -global Γ-spaces are trivial in degree 0 + (or at least weakly contractible) is crucial for the theorem, which is why we havebeen a bit more detailed in verifying that certain diagrams commute than usual.2.2.3.4. The Wirthm¨uller isomorphism.
Let α : H → G be an injective homo-morphism with ( G : im α ) < ∞ . If X is a genuine H -equivariant spectrum, the Wirthm¨uller isomorphism is a specific G -weak equivalence γ : α ! X → α ∗ X , seee.g. [ Hau17 , Proposition 3.7] or [
DHL + , Theorem 2.1.10].In a precise sense, the Wirthm¨uller isomorphism marks the distinction between genuine stable equivariant homotopy theory (encoding deloopings against all rep-resentation spheres) and na¨ıve stable equivariant homotopy theory (only admittingdeloopings against spheres with trivial actions). As non-equivariantly α ! is givenby a ( G : im α )-fold wedge, while α ∗ is a ( G : im α )-fold product, we can also viewthis as some sort of ‘twisted preadditivity.’ Below, we will establish an analogue ofthe Wirthm¨uller map γ for any H -global Γ-space X and prove: Theorem . Let α : H → G be an injective homomorphism such that ( G : im α ) < ∞ . Then the Wirthm¨uller map γ : α ! X → α ∗ X is a special G -globalweak equivalence for any X ∈ Γ- E M - H -SSet τ ∗ . The above theorem (in the guise of Corollary 2.3.6) will also be instrumentalin the proof of the equivalence between G -ultra-commutative monoids and special G -global Γ-spaces that we will give in the next section. Construction . Let us first construct the Wirthm¨uller map, whichactually already exists in the based context:Without loss of generality, we may assume that H is a subgroup of G and that α is its inclusion. Then α ! can be modeled by applying G + ∧ – : H -Set ∗ → G -Set ∗ levelwise and pulling through the E M -action. The counit is then given by i : X → G + ∧ H X, x [1 , x ]. Similarly, α ∗ is given by applying maps H ( G, –) levelwise.If now X is any pointed H -set, then we define the Wirthm¨uller map as the G -equivariant map γ : G + ∧ H X → maps H ( G, X ) adjunct to the H -equivariantmap G + ∧ H X → X, [ g, x ] ( g.x if g ∈ H ∗ otherwise.Explicitly, γ [ g, x ]( g ′ ) = ( g ′ g.x if g ′ g ∈ H ∗ otherwise . It is easy to check that the Wirthm¨uller map is natural; in particular, we can applyit levelwise to get a natural map γ : α ! Y → α ∗ Y for any Y ∈ Γ- E M - H -SSet τ ∗ . .2. G -GLOBAL Γ-SPACES 155 Just like we can think of Theorem 2.2.65 as twisted preadditivity, the basicidea of our proof will be similar to the proof of preadditivity (Theorem 2.2.60);however, the actual combinatorics are a bit more complicated and one has to beslightly careful in keeping track of all the actions involved.
Construction . If Y is a pointed G -set and X = res GH Y , then wecan partially ‘untwist’ the action on α ! X = G + ∧ H X by the usual G -equivariant shearing isomorphism shear : ( G/H ) + ∧ Y → G + ∧ H res GH Y, [[ g ] , y ] [ g, g − .y ]and dually we have a natural G -equivariant coshearing isomorphism coshear : Y G/H → maps H ( G, res GH Y )defined by coshear( y • )( g ) = g.y [ g − ] . Again, applying this levelwise we can extendthese to natural maps of G -global Γ-spaces. Construction . Let X be a G -global Γ-space. We define the twistedSegal map ̺ : X ( G/H + ∧ –) → maps H ( G, res GH X ) as the composition X ( G/H + ∧ –) ρ −→ X G/H coshear −−−−→ maps H ( G, res GH X )where ρ is obtained by applying in each degree the ‘generalized Segal map’ X ( G/H + ∧ S + ) → Q G/H X ( S + ) induced on factor [ g ] by p [ g ] ∧ S + : G/H + ∧ S + → S + .As in the previous section, ̺ is natural. Moreover, maps H ( G, res GH –) is ho-motopical in the G -global level weak equivalences by Corollary 2.2.46 and so is X ( G/H + ∧ –) for trivial reasons. As before we see that ̺ is also natural in mapsin the homotopy category. Lemma . Let T be a special G -global Γ -space. Then ̺ : T ( G/H + ∧ X ) → maps H ( G, res GH T ) is a G -global level weak equivalence. Proof.
It suffices to show that the generalized Segal map ρ : T ( G/H + ∧ –) → T G/H is a G -global level weak equivalence (where G acts via its action on T andon G/H ). For this we observe that for any finite set S the composition T (( G/H × S ) + ) ∼ = T ( G/H + ∧ S + ) ρ −→ T ( S + ) G/H ρ
G/HS −−−→ T (1 + ) G/H × S agrees with the Segal map ρ G/H × S for G/H × S , so it is a ( G × Σ G/H × S )-globalweak equivalence. Applying Corollary 2.2.45 to the homomorphism G × Σ S → G × Σ G/H × S induced by the identity of G and the homomorphism G × Σ S → Σ G/H × S classifying the obvious ( G × Σ S )-action on G/H × S , then shows that this is inparticular a ( G × Σ S )-global weak equivalence.Similarly, ρ S is a ( G × Σ S )-global weak equivalence, so ρ G/HS is a ( G × Σ S × Σ G/H )-global weak equivalence by Corollary 1.4.47, hence in particular a ( G × Σ S )-global weak equivalence. The claim now follows by 2-out-of-3. (cid:3) We can now use this to define a twisted version of the ‘multiplication map’ m T considered before: Construction . We write ν : G/H + → + for the map with ν [ g ] =1 for all g ∈ G , which induces for any X ∈ Γ- G - E M -SSet τ ∗ a natural map X ( ν ∧ –) : X ( G/H + ∧ –) → X . As before this descends to a natural map on thestrict homotopy category.
56 2. COHERENT COMMUTATIVITY
If now X = T is special, then we define n T : maps H ( G, res GH T ) → T as the mapin the strict homotopy category corresponding to the zig-zagmaps H ( G, res GH T ) T ( G/H + ∧ –) T. ∼ ̺ T ( ν ∧ –) Construction . We now define a ‘diagonal map’ δ : maps H ( G, X ) → maps H ( G, X )( G/H + ∧ –) for every H -global Γ-space X as follows: if S is a a finiteset, n ≥
0, and f : G → X ( S + ) n is H -equivariant, then δ ( f ) : G → X ( G/H + ∧ S + ) n is defined via δ ( f )( g ) = X ( i [ g − ] ∧ S + )( f ( g )), where i [ g ] : 1 + → S + sends 1 to [ g ]. Lemma . The above defines a map in Γ- E M - G -SSet τ ∗ . Proof.
We first show that δ ( f ) is H -equivariant, for which it is important toobserve that the H -action from the definition of maps H ( G, X )( G/H + ∧ S + ) is viathe H -action on X only; the H -action on G/H does not come into play yet.
Thus, h. (cid:0) δ ( f )( g ) (cid:1) = h. (cid:0) X ( i [ g − ] )( f ( g )) (cid:1) = X ( i [ g − ] ) (cid:0) h. ( f ( g )) (cid:1) = X ( i [ g − ] ) (cid:0) f ( hg ) (cid:1) = X ( i [( hg ) − ] ) (cid:0) f ( hg ) (cid:1) = δ ( f )( hg )where the second equality uses that X ( i [ g − ] ) is equivariant with respect to the H -action coming from X only , whereas the penultimate equation uses that [( hg ) − ] =[ g − h − ] = [ g − ] in G/H .Thus, δ indeed lands in maps H ( G, X )( G/H + ∧ –). It is then easy to check that δ is compatible with the simplicial structure maps, the structure maps of Γ, andthe E M -action, so it only remains to prove G -equivariance. For this we observethat the G -action on maps H ( G, X )( G/H + ∧ S + ) is via the diagonal of the right G -action on G and the left G -action on G/H . Thus, the chain of equalities δ ( g ′ .f )( g ) = X ( i [ g − ] ) (cid:0) ( g ′ .f )( g ) (cid:1) = X ( i [ g − ] ) (cid:0) f ( gg ′ ))= X (cid:0) ( g ′ . –) ∧ – (cid:1) X ( i [( gg ′ ) − ] ) (cid:0) f ( gg ′ ) (cid:1) = X (cid:0) ( g ′ . –) ∧ – (cid:1)(cid:0) δ ( f )( gg ′ ) (cid:1) precisely shows that δ is G -equivariant. (cid:3) Construction . Let us define for any pointed H -set X the H -equivariantmap j : X → res GH maps H ( G, X ) via j ( x )( g ) = ( g.x if g ∈ H ∗ otherwise . This is clearly natural and in particular we can apply it levelwise to get a naturalmap X → res GH maps H ( G, X ) for any H -global Γ-space. Proposition . Let F : maps H ( G, X ) → T be any morphism in the stricthomotopy category of Γ- G - E M -SSet τ ∗ . Then the diagram maps H ( G, res GH maps H ( G, X )) maps H ( G, res GH T )maps H ( G, X ) maps H ( G, X )( G/H + ∧ –) T ( G/H + ∧ –)maps H ( G, X ) T maps H ( G, res GH F )maps H ( G,j ) δ = F ( G/H + ∧ –) ̺ maps H ( G,X )( ν ∧ –) ̺T ( ν ∧ –) F commutes. .2. G -GLOBAL Γ-SPACES 157 Proof.
The right hand portion commutes by the above naturality considera-tions, and we will now prove that the two triangles on the left already commute onthe pointset level.Let us consider the lower triangle first. If S is a finite set, n ≥
0, and f : G → X ( S + ) is H -equivariant, thenmaps H ( G, X )( ν ∧ S + )( δ ( f ))( g ) = X ( ν ∧ S + )( δ ( f )( g ))= X ( ν ∧ S + ) X ( i [ g − ] ∧ S + ) (cid:0) f ( g )) = f ( g )for all g ∈ G , i.e. maps H ( G, X )( ν ∧ S + ) ◦ δ = id.Similarly, we compute for the upper left triangle ̺ ( δ ( f ))( g ) = coshear (cid:0) ρ ( δ ( f )) (cid:1) ( g ) = g. (cid:0) ρ ( δ ( f )) [ g − ] )= g. (cid:0) maps H ( G, X )( p [ g − ] ))( δ ( f )) (cid:1) ∈ res GH maps H ( G, X ( S + ) n ) , hence ̺ ( δ ( f ))( g )( g ) = (cid:0) g . (cid:0) maps H ( G, X )( p [ g − ] ))( δ ( f )) (cid:1)(cid:1) ( g )= (cid:0) maps H ( G, X )( p [ g − ] )( δ ( f )) (cid:1) ( g g )= X ( p [ g − ] ) (cid:0) δ ( f )( g g ) (cid:1) = X ( p [ g − ] ) X ( i [ g − g − ] ) (cid:0) f ( g g ) (cid:1) . If g ∈ H , then [ g − g − ] = [ g − ], hence X ( p [ g − ] ) X ( i [ g − g − ] ) = id. On the otherhand, if g / ∈ H , then [ g − g − ] = [ g − ] and X ( p [ g − ] ) X ( i [ g − g − ] ) factors throughthe base point. Thus,(2.2.15) ̺ ( δ ( f ))( g )( g ) = ( f ( g g ) if g ∈ H ∗ otherwise = ( g . (cid:0) f ( g ) (cid:1) if g ∈ H ∗ otherwisefor all g , g ∈ G , where the second equality uses H -equivariance of f .On the other hand, (cid:0) maps H ( G, j )( f ) (cid:1) ( g ) = j ( f ( g )) ∈ res GH maps H ( G, X ( S + ) n )for all g ∈ G , hence (cid:0) maps H ( G, j )( f ) (cid:1) ( g )( g ) = j ( f ( g ))( g ) = ( g . (cid:0) f ( g ) (cid:1) if g ∈ H ∗ otherwisefor all g , g ∈ G , which agrees with (2 . . (cid:3) Proof of Theorem 2.2.65.
By Corollary 2.2.45, G + ∧ H – and res GH are ho-motopical in the respective level equivalences, and they descend to an adjunc-tion between strict homotopy categories. We conclude that we have for every X, T ∈ Γ- G - I -SSet ∗ an isomorphism[ G + ∧ H X, T ] G → [ X, T ] H , F res GH ( F ) ◦ i ;here we write [ , ] H for the hom-sets in the strict homotopy category of H -globalΓ-spaces, and [ , ] G for the corresponding hom-sets of G -global Γ-sets. Using that γi = j , we are then reduced to showing that α : [maps H ( G, X ) , T ] → [ X, T ] H , f res GH ( F ) ◦ j is bijective whenever T is special.For this, we define an explicit inverse β : [ X, T ] H → [maps H ( G, X ) , T ] via β ( f ) = n T ◦ maps H ( G, f ); here we have used again that maps H ( G, –) descendsto strict homotopy categories.If now f : X → res GH T is any map in the strict homotopy category of H -globalΓ-spaces, then αβ ( f ) = α ( n T ◦ maps H ( G, f )) = res GH ( n T ) ◦ res GH (maps H ( G, f )) ◦ j =
58 2. COHERENT COMMUTATIVITY res GH ( n T ) ◦ j ◦ f by naturality of j . A straight-forward calculation then shows thatthe diagram res GH maps H ( G, res GH T )res GH T res GH ( T G/H )res GH T ( G/H ∧ –) j res GH k res GH ( T ( i [1] ∧ –)) res GH coshear ρ of H -equivariant maps commutes, where k : T → T G/H is the inclusion of the factorcorresponding to [1] ∈ G/H . From this we immediately conclude that res GH ( n T ) ◦ j =res GH (cid:0) T ( ν ∧ –) ◦ T ( i [1] ∧ –)) = id, hence αβ ( f ) = res GH ( n T ) ◦ j ◦ f = f as desired.Finally, βα ( F ) = β (res GH ( F ) ◦ j ) = n T ◦ maps H ( G, res GH F ) ◦ maps H ( G, j ), whichagrees with F by the previous proposition. (cid:3) A priori, the ultra-commutative or parsummable models are of a very differentnature than the models based on Γ-spaces. However, in this section we will con-struct specific homotopical functors ̥ : G -ParSumSSet → Γ- E M - G -SSet τ ∗ and ̥ : G -UCom → Γ- G - I -SSet τ ∗ and prove: Theorem . The diagram G -UCom G -ParSumSSetΓ- G - I -SSet special ∗ Γ- E M - G -SSet τ, special ∗ ̥ ev ω ̥ ev ω of homotopical functors comutes up to canonical isomorphism. Moreover, all thesefunctors induce equivalences of associated quasi-categories. Together with Theorem 2.1.46 this in particular shows that Schwede’s ultra-commutative monoids (i.e. commutative monoids for the box product on L -Top )are equivalent to a suitable notion of ‘special global Γ-spaces,’ connecting them toclassical approaches to equivariant coherent commutativity.On the other hand, together with Theorem 2.1.41 we can also view the aboveresult as a G -global strengthening of the equivalence between commutative monoidsfor the box product on I -SSet and E ∞ -monoids in SSet due to Sagave andSchlichtkrull [
SS12 , Theorem 1.2].Our construction of the functors ̥ (the archaic Greek letter digamma) is ananalogue of [ Sch19b , Construction 4.3] for so-called parsummable categories , whichwe will recall in Subsection 4.1.1. While Schwede used the letter γ , this is alreadytaken in our context by the Wirthm¨uller isomorphism (which plays a crucial rolein the proof of the theorem). Construction . Let us write Γ- G -ParSumSSet ∗ for the category of Set ∗ -enriched functors Γ → G -ParSumSSet , which we can identify as before withordinary functors X such that X (0 + ) is terminal. Then the evaluation functorev : Γ- G -ParSumSSet ∗ → G -ParSumSSet , X X (1 + ) has a left adjoint Fgiven by Set ∗ -enriched left Kan extension. .3. COMPARISON OF THE APPROACHES 159 Explicitly, (F X )( S + ) = X ⊠ S (as ⊠ is the coproduct on G -ParSumSSet ) withthe evident functoriality in X . The functoriality in S + is as follows: if f : S + → T + is any map in Γ, then (F X )( f ) is the map X ⊠ S → X ⊠ T given in each simplicialdegree by ( x s ) s ∈ S ( y t ) t ∈ T with y t = P s ∈ f − ( t ) x s . By direct inspection, theSegal maps X ⊠ S = (F X )( S + ) → X (1 + ) × S = X × S are precisely the inclusions, sothe underlying G -global Γ-space of F X is special by Theorem 2.1.13We now write ̥ for the composition(2.3.1) G -ParSumSSet F −→ Γ- G -ParSumSSet ∗ forget −−−→ Γ- E M - G -SSet τ ∗ The construction of ̥ : G -UCom → Γ- G - I -SSet τ ∗ is analogous.The main part of the proof of Theorem 2.3.1 will be establishing that thecomposition (2 . .
1) induces an equivalence of homotopy theories. To this end wewill introduce a suitable model structure on Γ- G -ParSumSSet ∗ and then showthat both F and forget are already equivalences of categories. On Γ- E M - G -SSet τ ∗ , we can define a boxproduct by performing the box product of E M -simplicial sets levelwise; this isindeed well-defined as ∗ ⊠ ∗ = ∗ . We can then identify Γ- G -ParSumSSet ∗ withthe category of commutative monoids for ⊠ on Γ- E M - G -SSet τ ∗ , which suggestsconstructing the required model structure via the general machinery recalled in2.1.3.1. While we cannot directly apply this (as ⊠ does not preserve initial ob-jects in each variable separately ), our arguments will still be very close to the usualapproach, and in particular we will need homotopical information about G -globalΓ-spaces of the form X ⊠ n / Σ n . Construction . Let X be a G -global Γ-space and let n ≥
1. The n -thsymmetric product SP n X is defined as X ⊠ n / Σ n ; we will confuse SP X with X .There is an evident way to make SP n into an endofunctor of Γ- E M - G -SSet ∗ .The map X ⊠ n → X ⊠ ( n +1) given by inserting the basepoint in the last factordescends to a natural map SP n X → SP n +1 X . We define the functor SP ∞ as thecolimit SP ⇒ SP ⇒ · · · along these natural maps. Remark . We have previously employed the notation Sym n X for X ⊗ n / Σ n for any closed symmetric monoidal model category C . Our reason for introducingnew notation is twofold: firstly, the notation Sym ∞ for SP ∞ would be ambiguous,as it is often used for ` n ≥ Sym n X ; secondly, the change of notation forces us toremember that we cannot apply any of the previous results on Sym n directly as ⊠ is not cocontinuous in each variable.The following theorem will be the key ingredient in establishing the modelstructure on Γ- G -ParSumSSet ∗ and comparing it to Γ- E M - G -SSet τ ∗ : Theorem . Let X ∈ Γ- E M - G -SSet τ ∗ and assume that X ( S + ) has no M -fixed points apart from the base point for all finite sets S . Then all the maps in X = SP X → SP X → · · · → SP ∞ X are G -global special weak equivalences. The proof of the theorem will be given below after some preparations. Webegin with the following consequences of the results of the previous sections:
60 2. COHERENT COMMUTATIVITY
Corollary . Let n ≥ and X ∈ Γ- E M - G -SSet τ ∗ . Then the maps X ∨ n → X ⊠ n ֒ → X × n (where the first map is induced by the n natural inclusions X → X ⊠ n ) are ( G × Σ n ) -global special weak equivalences. Proof.
Theorem 2.1.13 implies that the right hand map is even a (Σ n × G )-global level weak equivalence, so it suffices to prove the claim for the composition X ∨ n → X × n . For this we write Σ n ⊂ Σ n for the subgroup of those permutationsthat fix 1, and we let p : G × Σ n → G denote the projection. As in the proof ofCorollary 1.4.47 we have a ( G × Σ n )-equivariant isomorphismmaps G × Σ n ( G × Σ n , p ∗ X ) → X × n given on the i -th factor by evaluating at (1 , σ − i ), where σ i is any fixed permutationwith σ i (1) = i . Dually we have a ( G × Σ n )-equivariant isomorphism X ∨ n → ( G × Σ n ) + ∧ G × Σ n p ∗ X given on the i -th summand and in each simplicial degree by x [(1 , σ i ) , x ]. Onethen easily checks that the natural map X ∨ n → X × n factors as X ∨ n ∼ = −→ ( G × Σ n ) + ∧ G × Σ n p ∗ X γ −→ maps G × Σ n ( G × Σ n , p ∗ X ) ∼ = −→ X × n , so that the claim follows from Theorem 2.2.65. (cid:3) As we have already seen in several instances above, quotients by free groupactions are often fully homotopical, and in particular one can prove that quotieningout a free H -action sends ( G × H )-global special weak equivalences to G -globalspecial weak equivalences. Unfortunately, this does not yet imply Theorem 2.3.5;namely, while Σ n acts freely on X ⊠ n for any E M -simplicial set X without M -fixedpoints, in the situation of the theorem the Σ n -action is usually not free for n ≥ x , . . . , x n ) with 1 < k ≤ n base point components have non-trivial isotropy.However, this is the only thing that can go wrong: Lemma . Let X be a pointed E M -simplicial set without M -fixed pointsapart from the base point. Then the canonical Σ n -action on X ⊠ n is free outsidethose simplices with at least one base point component. Proof.
Let ( x , . . . , x n ) be an m -simplex such that no x i is the base point.We claim that x i = x j for all i = j , which will immediately imply that ( x , . . . , x n )has trivial isotropy.For the proof of the claim we let 1 ≤ i < j ≤ n be arbitrary. Then supp( x i ) = S mk =0 supp k ( x i ) is non-empty by assumption, so there exists a 0 ≤ k ≤ m withsupp k ( x i ) = ∅ . On the other hand, supp k ( x j ) ∩ supp k ( x i ) = ∅ = supp k ( x i ), hencesupp k ( x j ) = supp k ( x i ), and hence in particular x j = x i as desired. (cid:3) Using this, we can now salvage the above argument by exploiting the filtrationof X ⊠ n by the number of base point components. For this we will need the followingrelative version of the above ‘free quotient’ heuristic: Lemma . Let f : X → Y be a ( G × H ) -global special weak equivalencein Γ- E M -( G × H )-SSet τ ∗ . Assume that f is levelwise injective and that H actsfreely on Y ( S + ) outside the image of f for all finite sets S . Then f /H : X/H → Y /H is a G -global special weak equivalence. .3. COMPARISON OF THE APPROACHES 161 Proof.
We factor f as a ( G × H )-global special acyclic cofibration i followedby a fibration p (automatically acyclic). Proposition 2.2.54 implies that i/H is a G -global special weak equivalence, so to finish the proof it suffices that also p/H is.We claim that p/H is even a G -global level weak equivalence. For this welet K ⊂ M be any universal subgroup, S a finite K -set, and ϕ : K → G be ahomomorphism; we have to show that ( p ( S + ) /H ) ϕ = (cid:0) ( ϕ × H ) ∗ p ( S + ) /H (cid:1) K is aweak homotopy equivalence. Claim.
The map ( ϕ × H ) ∗ p ( S + ) is a ( K × H )-equivariant weak equivalence. Proof.
Let T ⊂ K × H be any subgroup. If T = Γ L,ψ for some L ⊂ K , ψ : L → H , then ( ϕ × H ) ∗ ( p )( S + ) T = p ( S + ) ( ϕ | L ,ψ ) is a weak homotopy equivalenceas p is an acyclic fibration in the positive special ( G × H )-global model structure,hence in particular a ( G × H )-global level weak equivalence.On the other hand, we claim that ( ϕ × H ) ∗ ( p )( S + ) T is even an isomorphismwhen T is not of the above form, for which it is enough that both ( ϕ × H ) ∗ ( f )( S + ) T and ( ϕ × H ) ∗ ( i )( S + ) T are isomorphisms.Indeed, ( ϕ × H ) ∗ ( f )( S + ) T is injective by assumption, so it only remains toshow that it is also surjective. But if any simplex of ( ϕ × H ) ∗ ( Y )( S + ) is fixed by T , then it is fixed by T ∩ H and this intersection is non-trivial by assumption. As H acts freely outside the image of f , we conclude that the simplex lies in the imageof f as desired.On the other hand, Corollary 2.2.26-(2) implies that also i ( S + ) is an injectivecofibration with free H -action outside the image, so the same argument shows that( ϕ × H ) ∗ ( i )( S + ) T is an isomorphism, completing the proof of the claim. △ As (–) /H : ( K × H )-SSet → K -SSet is left Quillen with respect to the A ℓℓ -model structures and since all objects are cofibrant in these, it follows immediatelythat ( ϕ × H ) ∗ ( p )( S + ) /H is a K -equivariant weak equivalence, which then impliesthe lemma by passing to K -fixed points. (cid:3) The filtration of X ⊠ n and SP n X according to the number of base point com-ponents is an instance of a more general construction for tensor powers which wewill now recall: Construction . Let C be a cocomplete symmetric monoidal category,and let f : X → Y be a morphism. We recall for each n ≥ n -cube C n and thefunctor K n ( f ) : C n → C from Construction 2.1.26.For 0 ≤ k ≤ n we let K nk ( f ) denote the subdiagram spanned by all those sets I with | I | ≤ k , and we define Q nk ( f ) := colim K kn ( f ). The inclusions of diagramshapes then induce X ⊗ n ∼ = Q n ( f ) i −→ Q n ( f ) i −→ · · · → Q nn − ( f ) i n −→ Q nn ( f ) ∼ = Y ⊗ n . where the outer isomorphisms are induced by the structure maps corresponding tothe unique terminal objects of K n and K nn , respectively.The composition of these is precisely f ⊗ n , while the composite map Q nn − ( f ) → Y ⊗ n was previously denoted f (cid:3) n .For any 1 ≤ k ≤ n there is a Σ n -action on Q nk ( f ) induced by the Σ n -action on K n and the symmetry isomorphisms of ⊗ . All of the above maps are Σ n -equivariant,and for X ⊗ n , Y ⊗ n and Q nn − ( f ) this recovers the actions considered before.
62 2. COHERENT COMMUTATIVITY
Theorem . Assume that in the above situation C is locally presentable and that ⊗ is cocontinuous in each variable. Moreover, let f : X → Y be any morphism in C and let ≤ k ≤ n . We write α : Σ n − k × Σ k → Σ n for the evident embedding. Then we have a Σ n -equivariant pushout square α ! ( X ⊗ ( n − k ) ⊗ Q kk − ( f )) α ! ( X ⊗ ( n − k ) ⊗ Y ⊗ k ) Q nk − Q nk ( f ) α ! ( X ⊗ ( n − k ) ⊗ f (cid:3) k ) i k and a pushout square Sym n − k X ⊗ ( Q kk − ( f ) / Σ k ) Sym n − k X ⊗ Sym k YQ nk − ( f ) / Σ n Q nk ( f ) / Σ n . Sym n − k X ⊗ ( f (cid:3) k / Σ k ) i k / Σ n Proof.
The assumptions guarantee that C is a closed symmetric monoidalmodel category in which all maps are cofibrations. The claim is therefore establishedin [ GG16 , proof of Theorem 22]. (In fact, going into their proof one only needsthe existence of pushouts and that the tensor product preserves these, but we wantto avoid repeating their argument.) (cid:3)
While the theorem as stated above does not directly apply to the levelwisebox product on Γ- E M - G -SSet τ ∗ , we can apply it to the levelwise box producton Γ- E M - G -SSet τ . As the full subcategory Γ- E M - G -SSet τ ∗ is closed under allconnected colimits, we then see a posteriori that we also have the correspondingpushouts there.With this established, we can adapt the proof strategy of [ GG16 , Corollary 23]to deduce Theorem 2.3.5 from the Wirthm¨uller isomorphism:
Proof of Theorem 2.3.5.
We will prove that the composite X → SP n X (induced by any of the n canonical embeddings X → X ⊠ n ) is a G -global weakequivalence for any n ≥
1. By 2-out-of-3 we can then conclude that SP n X → SP n +1 X is a G -global special weak equivalence, and so is X → SP ∞ X as transfinitecomposition of G -global special weak equivalences.It remains to prove the claim. For this we let f : ∗ → X be the inclusion of thebasepoint. We will now prove more generally by induction on n :(1) For all 2 ≤ k ≤ n and any group H the map i k : Q nk − ( f ) → Q nk ( f ) is a( G × H × Σ n )-global special weak equivalence when we let H act trivially.(2) For all 2 ≤ k ≤ n the induced map i k / Σ n is a G -global weak equivalence.For n = 1 there is nothing to prove, so assume n ≥
2. If 2 ≤ k < n , then we let α : Σ n − k × Σ k → Σ n be the evident embedding again. By the induction hypothesis, f (cid:3) k is in particular a ( G × H × Σ n − k × Σ k )-global special weak equivalence for anygroup H . Moreover, it is an injective cofibration by Corollary 2.1.38 (where we haveagain used that we can form the corresponding colimit in Γ- E M - G -SSet τ ), hencealso α ! ( f (cid:3) k ) is an injective cofibration. On the other hand, Proposition 2.2.54 showsthat it is a ( G × H × Σ n )-global special weak equivalence. Applying Lemma 2.2.53 .3. COMPARISON OF THE APPROACHES 163 to the pushout α ! Q kk − ( f ) α ! ( X ⊠ k ) Q nk − ( f ) Q nk ( f ) i k from the above discussion therefore shows that i k is a ( G × H )-global special weakequivalence as desired.Similarly, we deduce part (2) for 2 ≤ k < n from the induction hypothesistogether with the pushout Q kk − ( f ) / Σ k SP k XQ nk − ( f ) / Σ n Q nk ( f ) / Σ n , so it only remains to treat the case k = n of both statements.For the first statement we observe that the composition X ∨ n ∼ = Q n ( f ) → Q n ( f ) → · · · Q nn − ( f ) → Q nn ( f ) ∼ = X ⊠ n , where the unlabelled isomorphism is induced on the i -th summand by the iteratedunitality isomorphism X ∼ = K n ( { i } ), is precisely the canonical map X ∨ n → X ⊠ n ,hence a ( G × H × Σ n )-global special weak equivalence by Corollary 2.3.6 appliedto X with trivial H -action. On the other hand, we have seen above that all themaps in Q n ( f ) → Q n ( f ) → · · · → Q nn − ( f ) are ( G × H × Σ n )-global special weakequivalences, so also Q nn − ( f ) → Q nn ( f ) is a ( G × H × Σ n )-global special weakequivalence by 2-out-of-3.For the second statement, we observe that i n is a ( G × Σ n )-global weak equiva-lence by the first statement, and an injective cofibration as seen above. On the otherhand, the image of i n ( S + )( ω ) : Q nn − ( f )( S + )( ω ) → X ⊠ n ( S + )( ω ) ∼ = X ( S + )( ω ) ⊠ n contains all n -tuples of simplices with at least one basepoint component by con-struction, so the Σ n -action is free outside the image of i n ( S + )( ω ) by Lemma 2.3.7.We may therefore conclude from Lemma 2.3.8 that also i n / Σ n is a G -global weakequivalence, which completes the proof of the theorem. (cid:3) In this section we will establish the modelstructure on Γ- G -ParSumSSet ∗ and compare it to Γ- E M - G -SSet τ ∗ . Construction . If X ∈ Γ- E M - G -SSet τ ∗ , then SP ∞ X becomes natu-rally an element of Γ- G -ParSumSSet ∗ by declaring for each finite set S that theunit of (SP ∞ X )( S + ) should be the basepoint and that the composition should beinduced by the canonical isomorphisms X ⊠ m ⊠ X ⊠ n ∼ = X ⊠ ( m + n ) . We omit the easyverification that this lifts SP ∞ to Γ- E M - G -SSet τ ∗ → Γ- G -ParSumSSet ∗ .The structure map of SP X yields a natural map η : X → forget SP ∞ X , andwe have for each Y ∈ Γ- G -ParSumSSet ∗ a natural map ǫ : SP ∞ (forget Y ) → Y induced by the iterated multiplication maps Y ⊠ n → Y . We omit the easyverification that η and ǫ satisfy the triangle identities, making them into unit andcounit, respectively, of an adjunction SP ∞ ⊣ forget.
64 2. COHERENT COMMUTATIVITY
Theorem . There exists a unique model structure on Γ- G -ParSumSSet ∗ in which a map is a weak equivalence or fibration if and only if it so in the special G -global model structure Γ- E M - G -SSet τ ∗ . Moreover, the adjunction SP ∞ : Γ- E M - G -SSet τ ∗ ⇄ Γ- G -ParSumSSet ∗ : forget is a Quillen equivalence. One might be tempted to prove the theorem by first constructing a suitablelevel model structure and then Bousfield localizing. However, in this approach itis not clear a priori how the resulting weak equivalences in Γ- G -ParSumSSet ∗ relate to the special G -global weak equivalences of underlying G -global Γ-spaces.Instead, the main ingredient to our proof will be Theorem 2.3.5 established above.However, there are some pointset level issues we will have to deal with first. Lemma . Let X be any G -global Γ -space. Then X ⊠ – preserves G -globalspecial weak equivalences. Proof.
The functor X × – preserves G -global special weak equivalences byCorollary 2.2.63. If now Y is any G -global Γ-space, then the inclusion X ⊠ Y ֒ → X × Y is a G -global level weak equivalence by Theorem 2.1.11. Moreover, it isclearly natural, so the claim follows by 2-out-of-3. (cid:3) Proposition . Let f : X → Y be a levelwise injective G -global spe-cial weak equivalence in Γ- G - E M -SSet τ ∗ such that the M -actions on X ( S + ) and Y ( S + ) have no fixed points apart from the base point for all finite sets S .Then f (cid:3) n / Σ n is a G -global special weak equivalence. Proof.
We first observe that Sym n f = SP n f is a G -global special weak equiv-alence by Theorem 2.3.5 together with 2-out-of-3.With this established, we can prove the proposition by induction on n similarlyto the above. For n = 1 the claim is trivial; for n ≥ Q n ( f ) / Σ n → Q n ( f ) / Σ n → · · · → Q nn − ( f ) → Q nn ( f ) / Σ n are G -global special weak equivalences; the claim will then follow because the righthand map is conjugate to f (cid:3) n / Σ n .For the proof of the claim we observe that Q nk − ( f ) / Σ n → Q nk ( f ) / Σ n is apushout of SP n − k X ⊠ ( f (cid:3) k / Σ k ) as seen in the previous section. For k < n , this isa G -global special weak equivalence by the induction hypothesis together with theprevious lemma. As it is moreover levelwise injective by Corollary 2.1.38 togetherwith Lemma 2.1.37, the claim follows immediately. It only remains to consider thecase k = n , for which we observe that the composition (2 . .
2) is conjugate to SP n f ,hence a G -global special weak equivalence by the above observation. The claimtherefore follows by 2-out-of-3. (cid:3) Proposition . Let f : A → B be a map in Γ- E M - G -SSet τ ∗ , and let (2.3.3) SP ∞ A SP ∞ BX Y SP ∞ fg .3. COMPARISON OF THE APPROACHES 165 be a pushout in Γ- G -ParSumSSet . Then the underlying map forget g of G -global Γ -spaces can be written as a transfinite composition X = Y g −→ Y g −→ Y → · · · → Y ∞ = Y where each g n fits into a pushout square (2.3.4) X ⊠ Q nn − ( f ) / Σ n X ⊠ SP n ( B ) Y n − Y nX ⊠ f (cid:3) n / Σ n g n in Γ- E M - G -SSet τ ∗ . The definition of the vertical maps in (2 . .
4) is complicated, but we will fortu-nately never need their explicit form.
Proof.
We recall the left adjoint P : Γ- E M - G -SSet τ → Γ- G -ParSumSSet of the forgetful functor; explicitly, P X = ` n ≥ Sym n X with the evident functo-riality in X . The unit is given by X ∼ = Sym X ֒ → P X . There is a natural map P X → SP ∞ X induced by the structure maps Sym n X = SP n X → SP ∞ X . Observethat p is compatible with the adjunction units in the sense that pη = η . Claim.
The naturality square P A P B SP ∞ A SP ∞ B P fp p SP ∞ f is a pushout in Γ- G -ParSumSSet . Proof.
We first observe that the pushout of X ← P A → P B formed in Γ- G -ParSumSSet belongs to Γ- G -ParSumSSet ∗ as ( P f )(0 + ) = P ( f (0 + )) is anisomorphism because f (0 + ) is. It therefore suffices to check the universal propertywith respect to every T ∈ Γ- G -ParSumSSet ∗ , which is an easy diagram chaseusing the defining properties of P and SP ∞ as left adjoints and the compatibilityof p with the unit maps. △ As the inclusion of Γ- G -ParSumSSet ∗ preserves connected colimits (hence inparticular pushouts), (2 . .
3) is also a pushout in Γ- G -ParSumSSet . Pasting withthe pushout from the claim we therefore get a pushout P A P BX Y P fg in Γ- G -ParSumSSet . As Γ- E M - G -SSet τ becomes a symmetric monoidal modelcategory with respect to the levelwise box product when we declare all maps tobe both cofibrations and fibrations, [ Whi17 , Proposition B.2] shows that as a
66 2. COHERENT COMMUTATIVITY morphism in Γ- E M - G -SSet τ , g can be written as a transfinite composition Y g −→ Y → · · · with each g n fitting into a pushout(2.3.5) X ⊠ Q nn − ( f ) / Σ n X ⊠ Sym n ( B ) Y n − Y nX ⊠ f (cid:3) n / Σ n g n in E M - G -SSet τ . Note that it does not matter whether we form Q nn − ( f ) / Σ n and f (cid:3) n / Σ n in E M - G -SSet ∗ or in E M - G -SSet as the former is closed underconnected colimits. To finish the proof it therefore suffices that Y n belongs to Γ- E M - G -SSet τ ∗ and that (2 . .
5) is a pushout in Γ- E M - G -SSet τ ∗ for all n ≥ (cid:3) Proof of Theorem 2.3.12.
Let us first prove the existence of the modelstructure, for which we will use Crans’ Transfer Criterion (Proposition 1.1.5).The inclusion i : Γ- G -ParSumSSet ∗ ֒ → Γ- G -ParSumSSet preserves filteredcolimits, and it admits a left adjoint q given by q ( X )( S + ) = X ( S + ) /X (0 + ), so Γ- G -ParSumSSet ∗ is locally presentable as an accessible Bousfield localizationof a locally presentable category. In particular, any set of maps permits the smallobject argument, and we therefore only have to show that every relative SP ∞ ( J )-cellcomplex is a weak equivalence for a suitable set J of generating acyclic cofibrationsof the positive special model structure on Γ- E M - G -SSet τ ∗ .For this we pick any such set J all of whose maps have cofibrant sources;this is possible by Theorem 2.2.51. As filtered colimits and weak equivalences in Γ- G -ParSumSSet ∗ are created in Γ- E M - G -SSet τ ∗ and since filtered colimits inthe latter are homotopical, it suffices that in any pushoutSP ∞ A SP ∞ BX Y SP ∞ jg with j ∈ J also g is a weak equivalence in Γ- G -ParSumSSet ∗ .By Lemma 2.3.13 together with Proposition 2.3.14 we see that X ⊠ j (cid:3) n / Σ n isa G -global special weak equivalence for every n ≥
0. As it is moreover an injectivecofibration by the same argument as above, also every pushout of such a map is a G -global special weak equivalence by Lemma 2.2.53. We therefore conclude fromthe previous proposition that g can be written as a transfinite composition of weakequivalences, so it is a weak equivalence itself, completing the verification of Crans’criterion and hence of the existence of the model structure.The forgetful functor preserves and reflects weak equivalences by definition;to see that SP ∞ ⊣ forget is a Quillen equivalence it therefore only remains toshow that the unit X → forget SP ∞ X is a weak equivalence for every cofibrant X ∈ Γ- E M - G -SSet τ ∗ . But for any such X , the M -action on X ( S + ) has no M -fixed points apart from the basepoint for any finite set S by Corollary 2.2.26-(1);the claim is therefore an instance of Theorem 2.3.5. (cid:3) In this section we will completethe proof of Theorem 2.3.1 by comparing Γ- G -ParSumSSet ∗ to G -ParSumSSet . .3. COMPARISON OF THE APPROACHES 167 Proposition . The adjunction (2.3.6) F : G -ParSumSSet ⇄ Γ- G -ParSumSSet ∗ : ev is a Quillen equivalence with fully homotopical left adjoint. Proof.
The (acyclic) fibrations in Γ- G -ParSumSSet ∗ are created in the G -global special model structure on Γ- E M - G -SSet τ ∗ , hence they are in particular(acyclic) fibrations in the G -global level model structure. On the other hand, the(acyclic) fibrations of G -ParSumSSet are created in E M - G -SSet τ ∗ , so it imme-diately follows that ev is right Quillen, i.e. (2 . .
6) is a Quillen adjunction.Next, we observe that ev is homotopical in G -global level weak equivalences,and hence in particular homotopical in weak equivalences between special Γ- G -parsummable simplicial sets by Lemma 2.2.52. As any fibrant object of the abovemodel structure on Γ- G -ParSumSSet ∗ is special, we conclude that R ev can becomputed by taking a special replacement.On the other hand, Corollary 2.1.14 shows that F is homotopical, and we havealready noted in Construction 2.3.2 that it takes values in special Γ- G -parsummablesimplicial sets. Together with the above we conclude that the ordinary unit X → ev(F X ) already represents the derived unit for all G -parsummable simplicial sets;this shows that the derived unit is an isomorphism. Finally, Lemma 2.2.52 showsthat R ev is conservative; it follows that Ho(F) ⊣ R ev is an adjoint equivalence,finishing the proof of the proposition. (cid:3) Proof of Theorem 2.3.1.
The functor ev ω : I -SSet → E M -SSet is strongsymmetric monoidal by Proposition 2.1.20, so ev ω : UCom → ParSumSSet pre-serves finite coproducts. It follows easily that the canonical mate F ◦ ev ω ⇒ ev ω ◦ Fof the identity transformation Γ- G -UCom ∗ Γ- G -ParSumSSet ∗ G -UCom G -ParSumSSet ev ω ev ev ⇒ ev ω is an isomorphism. On the other hand, ev ω commutes with the forgetful functorson the nose, so we altogether get the desired isomorphism filling G -UCom G -ParSumSSetΓ- G - I -SSet ∗ Γ- E M - G -SSet τ ∗ . ̥ ev ω ̥ ev ω The horizontal maps in this induce equivalences on quasi-categories by Corol-lary 2.1.36 and Theorem 2.2.29, respectively. Moreover, the right hand verticalarrow induces an equivalence by Theorem 2.3.12 together with the previous propo-sition. By 2-out-of-3 we conclude that also the left hand vertical arrow descends toan equivalence of quasi-categories, which completes the proof of the theorem. (cid:3)
Together with Theorem 2.1.46 we immediately conclude:
Corollary . There exists a preferred equivalence between: • the quasi-category of ultra-commutative monoids in the sense of [ Sch18 ]with respect to F in -global weak equivalences (cf. Subsection 2.1.6), and
68 2. COHERENT COMMUTATIVITY • the quasi-category of special Γ - E M -spaces with respect to the global levelweak equivalences. (cid:3) On the other hand, Theorem 2.3.1 also provides information about genuineequivariant infinite loop spaces:
Theorem . There exists an essentially preferred equivalence between • the quasi-category ( Γ- G -SSet special ∗ ) ∞ of special Γ - G -spaces (with respectto the G -equivariant level weak equivalences), and • the quasi-category G -UCom ∞ of G -ultra-commutative monoids with re-spect to those maps f such that u G f is a G -equivariant weak equivalence. As mentioned in the introduction of this section, the special case G = 1 wasknown by [ SS12 , Theorem 1.2] and the usual comparison between E ∞ -algebras andΓ-spaces. It seems that the above generalization has not appeared in the literaturebefore, even for finite G . Proof.
By Theorem 2.3.1 and its proof, ̥ restricts to a functor G -UCom → Γ- G - I -SSet special ∗ that descends to an equivalence of quasi-localizations with re-spect to the G -global weak equivalences and G -global level weak equivalences, re-spectively. In particular, the composition with ev U G exhibits the special Γ- G -spacesas a (Bousfield) localization of G -UCom by Theorem 2.2.55.It only remains to show that this composition precisely inverts the u G -weakequivalences. But indeed, a map g : X → Y of special Γ- G -spaces is a G -global levelweak equivalence if and only if g (1 + ) is an ordinary G -equivariant weak equivalence;the claim follows as ev U G ◦ ̥ ( f ) is conjugate to ev U G ( f (1 + )) for any map f of G -global Γ-spaces. (cid:3) HAPTER 3
Stable G -global homotopy theory In this chapter we introduce stable G -global homotopy theory as a joint gen-eralization of G -equivariant stable homotopy theory and global stable homotopytheory (with respect to finite groups) in the sense of Schwede [ Sch18 , Chapter 4].We then discuss several connections to the models discussed in the previous twochapters, and in particular we will prove a G -global version of Segal’s classical De-looping Theorem , relating G -global spectra to G -globally coherently commutativemonoids. G -global homotopy theory of G -spectra3.1.1. Recollections on (equivariant) stable homotopy theory. We be-gin by recalling symmetric spectra [ HSS00 ] which will serve as the basis of ourmodels of G -global homotopy theory below. Construction . We write Σ for the following SSet ∗ -enriched category:the objects of Σ are precisely the finite sets. If A and B are finite sets, thenmaps Σ ( A, B ) = _ i : A → B injective S B r i ( A ) . If C is yet another finite set, then the composition maps Σ ( A, B ) ∧ maps Σ ( B, C ) → maps( A, C ) is given on the summands corresponding to i : A → B , j : B → C by S B r i ( A ) ∧ S C r j ( B ) → S j ( B ) r ji ( A ) ∧ S C r j ( B ) ∼ = S C r ji ( A ) ֒ → maps Σ ( A, C )where the unlabelled arrow on the left is induced by j , the isomorphism is thecanonical one, and the final map is the inclusion of the summand corresponding tothe injection ji . Definition . A symmetric spectrum (or, by slight abuse of language,‘spectrum’ for short) is a SSet ∗ -enriched functor Σ → SSet ∗ . We write Spectra for the
SSet ∗ -enriched functor category Fun ( Σ , SSet ∗ ). Definition . Let G be any discrete group, possibly infinite. A G -spectrum (or, more precisely, a G -symmetric spectrum ) is a G -object in Spectra . We write G -Spectra for the SSet ∗ -enriched category of G -spectra.For finite G , Hausmann [ Hau17 ] studied G -spectra as a model of G -equivariantstable homotopy theory. In the rest of this subsection we will recall some of hisresults, as the G -global theory sometimes parallels the equivariant ones. We restrictourselves to the foundations here and will recall other results later as needed. Remark . Strictly speaking, [
HSS00 ] and [
Hau17 ] define symmetricspectra as a sequence of based Σ n -simplicial sets X n , n ≥
0, together with suitablyequivariant and associative structure map S m ∧ X n → X m + n ; the equivalence to G -GLOBAL HOMOTOPY THEORY the above definition is noted for example in [ Hau17 , 2.4]. In what follows we willtacitly translate the results of [
Hau17 ] to the above language where necessary.3.1.1.1.
Equivariant level model structures.
Before we can introduce the sta-ble homotopy theory of G -spectra, we first need to consider suitable level modelstructures. Proposition . There exists a (unique) model structure on G -Spectra in which a map f is weak equivalence or fibration if and only if f ( A ) is a G G, Σ A -weak equivalence or fibration, respectively, for every finite set A . We call this the G -equivariant projective level model structure . It is proper, combinatorial, andsimplicial. Proof. [ Hau17 , Corollary 2.26] and the discussion following it show that themodel structure exists, that it is proper, and cofibrantly generated (hence combi-natorial). Finally, as all the relevant constructions are levelwise, one immediatelychecks that the cotensoring is a right Quillen bifunctor, i.e. the above is a simplicialmodel category. (cid:3)
To describe the generating cofibrations we introduce the following notation:
Construction . Let A be any finite set. Then the evaluation functorev A : Spectra → Σ A -SSet ∗ admits a left adjoint G A by the Special Adjoint Func-tor Theorem (or, alternatively, as a SSet ∗ -enriched left Kan extension). If X is anypointed Σ A -simplicial set, then we call the spectrum G A X a semi-free spectrum ;explicitly, ( G A X )( B ) = Σ ( A, B ) ∧ Σ A X .As the SSet ∗ -functor ev A preserves cotensors, it follows formally that there isa unique SSet ∗ -enrichment on G A such that G A ⊣ ev A is a SSet ∗ -enriched adjunc-tion, and that with respect to this enrichment G A preserves tensors. If now G is anydiscrete group, then the evaluation functor ev A : G -Spectra → ( G × Σ A )-SSet ∗ admits a (simplicial or SSet ∗ -enriched) left adjoint by applying the functor G A considered before and pulling through the G -action via functoriality.An easy adjointness argument then shows that the maps G A (( G × Σ A ) /H + ∧ ∂ ∆ n ֒ → ∆ n ) + ) for H ∈ G G, Σ A form a set of generating cofibrations, and similarlyfor the generating acyclic cofibrations.We will be mostly interested in a variant of the above model structure that hasmore cofibrations. Definition . A map f : X → Y of spectra is called a flat cofibration if ithas the left lifting property against those p : S → T such that p ( A ) : S ( A ) → T ( A )is a Σ A -equivariant acyclic fibration for all finite sets A .If G is any group, then a map f : X → Y of G -spectra is called a flat cofibration if its underlying map of non-equivariant spectra is a flat cofibration in the abovesense. A G -spectrum X is called flat if 0 → X is a flat cofibration. Example . The sphere spectrum S is flat (since it is even projectivelycofibrant), and hence in particular flat as a G -spectrum (equipped with the trivialor in fact any G -action) for any group G . More generally, if T is any set, then W T S is flat since coproducts in any model category preserve cofibrant objects. Example . Let T be a finite set. Then it is a non-trivial result that also Q T S is flat, see [ Ost16 , Proposition B.1]. .1. G -GLOBAL HOMOTOPY THEORY OF G -SPECTRA 171 Proposition . There exists a (unique) model structure on G -Spectra in which a map f is weak equivalence or fibration if and only if f ( A ) is a weak equiv-alence or fibration, respectively, in the injective G G, Σ A -equivariant model structurefor every finite set A . The cofibrations of this model category are precisely the flatcofibrations. We call this the G -equivariant flat level model structure . It is proper,combinatorial, and simplicial. Proof. [ Hau17 , Corollary 2.25] constructs a model structure with the de-sired weak equivalences and fibrations, and shows that it is proper and cofibrantlygenerated (hence combinatorial). The same argument as above shows that thismodel structure is simplicial. It remains to show that the resulting cofibrationsare precisely the flat cofibrations. This appears as [
Hau17 , Remark 2.20], but wedetail the argument here as we will need it again later.We first observe that the model structure can be constructed as an instanceof Schwede’s criterion (see Proposition 1.3.5); the consistency condition is verifiedas [
Hau17 , Proposition 2.24]. In particular, the cofibrations are those maps f such that each latching map (cf. Subsection 1.3.3) is a cofibration in the injective G G, Σ A -equivariant model structure, i.e. a cofibration of underlying simplicial sets.As the latching objects are independent of the the group G (they are definedin terms of left Kan extensions, which are compatible with passing to functor cat-egories), this shows that a map in G -Spectra is a cofibration in the above modelstructure if and only if it is so in . But the latter are by definitionprecisely the flat cofibrations, which completes the proof of the proposition. (cid:3) Equivariant stable homotopy theory.
The main disadvantage of sym-metric spectra is that the correct notion of ‘stable weak equivalence’ turns out tobe coarser than those maps inducing isomorphisms on the na¨ıvely defined homotopygroups—even worse, the correct weak equivalences are only obtained indirectly byBousfield localizing at the Ω -spectra . The equivariant situation is analogous:
Definition . A G -spectrum X is called a G - Ω -spectrum if the followingholds: for all H ⊂ G and all finite H -sets A, B , the adjoint structure map X ( A ) → R Ω B X ( A ∐ B ) is an H -equivariant weak equivalence. Here H acts on both sidesvia its actions on X and on A and B (hence on A ∐ B ). Theorem . There is a unique model structure on G -Spectra whosecofibrations are the G -equivariant projective cofibrations, and whose fibrant objectsare those G - Ω -spectra that are fibrant in the G -equivariant projective level modelstructure. This model structure is simplicial, proper, and combinatorial. We callit the G -equivariant stable projective model structure , or G -equivariant projectivemodel structure for short.Similarly, there is a unique model structure on G -Spectra whose cofibrationsare the flat cofibrations, and whose fibrant objects are those G - Ω -spectra that arefibrant in the G -equivariant flat level model structure. This model structure is againsimplicial, proper, and combinatorial. We call it the G -equivariant stable flat modelstructure , or G -equivariant flat model structure for short. Proof.
Proper combinatorial model structures with the corresponding cofibra-tions and fibrant objects are constructed as [
Hau17 , Theorem 4.8] and [
Hau17 ,Theorem 4.7], respectively, and this completely determines the model structures byProposition A.2.7.
72 3. STABLE G -GLOBAL HOMOTOPY THEORY It only remains to show that these model structures are simplicial, for which itsuffices (by the corresponding statements for the level model structures) that thepushout product i (cid:3) j of a cofibration i of a simplicial sets with a G -equivariantacyclic cofibration is a G -equivariant acyclic cofibration again, which appears as[ Hau17 , Lemma 4.5]. (cid:3)
Na¨ıve homotopy groups.
As before, we can by abstract nonsense char-acterize the weak equivalences of the above model structures as the maps f : X → Y such that [ f, T ] : [ Y, T ] → [ X, T ] is an isomorphism for any G -Ω-spectrum T , where[ , ] denotes the hom sets in the homotopy category with respect to the G -equivariantlevel weak equivalences. In particular, the two model structures have the same weakequivalences, which we call the G -equivariant (stable) weak equivalences .As in the non-equivariant setting, these are quite hard to grasp. On the otherhand, it is often easier to show that a map f : X → Y of ordinary symmetric spectrainduces an isomorphism of the na¨ıve homotopy groups, and any such map is indeeda stable weak equivalence by [ HSS00 , Theorem 3.1.11]. This motivates looking fora generalization of the stable homotopy groups to the equivariant setting:
Construction . Let U be a complete H -set universe and let Y be any H -spectrum (e.g. the underlying H -spectrum of a G -spectrum X with H ⊂ G ).We define(3.1.1) π U k ( Y ) = colim A ∈ s ( U ) [ S A ∐{ ,...,k } , X ( A )] H ∗ for k ≥ π U k ( Y ) = colim A ∈ s ( U ) [ S A , X ( A ∐ { , . . . , − k } )] H ∗ for k <
0, where s ( U ) is the filtered poset of finite H -subsets of U and [– , –] H ∗ denotesthe set of maps in the based H -equivariant homotopy category. The structure mapof (3 . .
1) for an inclusion A ⊂ B is given by[ S A ∐{ ,...,k } , X ( A )] H ∗ S B r A – −−−−→ [ S B ∐{ ,...,k } , S B r A ∧ X ( A )] ∗ σ −→ [ S B ∐{ ,...,k } , X ( B )] , where we have secretly identified S B r A ∧ S A ∐{ ,...,k } ∼ = S B ∐{ ,...,k } via the obviousisomorphism. The definition of the structure maps for k ≤ H once and for all acomplete H -set universe U H , and abbreviate π H ∗ := π U H ∗ . Definition . A map f : X → Y of G -spectra is called a π ∗ -isomorphism if π H ∗ f is an isomorphism of Z -graded abelian groups for every subgroup H ⊂ G .Analogously to the non-equivariant situation we have: Theorem . Any π ∗ -isomorphism of G -spectra is a G -equivariant stableweak equivalence. Proof.
See [
Hau17 , Theorem 3.26]. (cid:3)
Remark . Any two complete H -set universes U , U ′ are isomorphic, andany choice of such isomorphism yields a natural isomorphism π U∗ ∼ = π U ′ ∗ ; in partic-ular, the notion of π ∗ -isomorphism is independent of any choices. However, thisisomorphism is not canonical, and in a precise sense this non-canonicity capturesthe failure of π ∗ to compute the ‘true’ homotopy groups, see [ Hau17 , 3.3–3.4]. .1. G -GLOBAL HOMOTOPY THEORY OF G -SPECTRA 173 Stability.
The stable model structures from Theorem 3.1.12 are indeedstable in the model categorical sense, i.e. the suspension/loop adjunction on thehomotopy category is an equivalence. We will need the following pointset levelstrengthening of this later:
Proposition . The adjunction
Σ = S ∧ – : G -Spectra G -equiv. proj. ⇄ G -Spectra G -equiv. proj. : maps( S , –) = Ω is a Quillen equivalence. Moreover: (1) Σ preserves G -equivariant weak equivalences. (2) Ω preserves G -equivariant weak equivalences between G -spectra that arefibrant in the G -equivariant projective level model structure. Proof.
The above adjunction is a Quillen adjunction as the G -equivariantprojective model structure is simplicial. Moreover, S ∧ – obviously preserves G -equivariant level weak equivalences; as any G -equivariant weak equivalence factorsas an acyclic cofibration followed by an G -equivariant level weak equivalence, thisshows that Σ is in fact homotopical.To see that it is a Quillen equivalence, we consider the category G -Spectra Top of symmetric G -spectra in topological spaces, i.e. G -objects in Fun ( Σ , Top ∗ ). Thiscarries a ‘ G -equivariant projective’ model structure [ Hau17 , Theorem 4.8] suchthat the adjunction | – | ⊣ Sing preserves and reflects weak equivalences [
Hau17 ,Proposition 2.38]. As unit and counit of this are even G -equivariant level weakequivalences, we conclude that | – | ⊣ Sing induces an equivalence between the cor-responding homotopy categories. Now consider the diagram G -Spectra G -Spectra G -Spectra Top G -Spectra Top | – | S ∧ – | – | S ∧ – of homotopical functors, commuting up to natural isomorphism. By the above, thevertical arrows induce equivalences of homotopy categories, and so does the lowerarrow by [ Hau17 , Proposition 4.9]. By 2-out-of-3, also the top horizontal arrowdescends to an equivalence, i.e. Σ ⊣ Ω is a Quillen equivalence.It remains to show that Ω sends any G -equivariant weak equivalence f : X → Y of level fibrant G -spectra to a G -equivariant weak equivalence. For this we considerthe naturality square X Sing | X | Y Sing | Y | ; ηf Sing | f | η as the horizontal arrows are G -equivariant level weak equivalences, Sing | f | is a G -equivariant weak equivalence. On the other hand, Sing Z is obviously level fibrantfor any Z ∈ G -Spectra Top ; as Ω preserves G -equivariant level weak equivalencesbetween level fibrant G -spectra, it therefore suffices to show that Ω Sing | f | , whichis conjugate to Sing Ω | f | , is a G -equivariant weak equivalences. But indeed, | f | is a G -equivariant weak equivalence, so Ω | f | is a G -equivariant weak equivalenceby [ Hau17 , Proposition 4.2-(3)], and hence so is Sing Ω | f | as desired. (cid:3)
74 3. STABLE G -GLOBAL HOMOTOPY THEORY G -global model structures. In [
Hau19b , Theorem 2.18], Hausmannintroduced a global model structure on the category
Spectra of ordinary symmetricspectra analogous to Schwede’s global model structure on orthogonal spaces [
Sch18 ,Theorem 4.3.17]. We will now generalize this construction to yield G -global modelstructures on G -Spectra for any discrete G ; while we cast them differently, ourarguments for this are mostly analogous to Hausmann’s.3.1.2.1. Level model structures.
Before we can construct the desired modelstructures on G -Spectra modelling stable G -global homotopy theory, we againneed to introduce several models with a stricter notion of weak equivalence: Proposition . There is a (unique) model structure on G -Spectra inwhich a map f : X → Y is a weak equivalence or fibration if and only if for everyfinite set A the (Σ A × G ) -equivariant map f ( A ) : X ( A ) → Y ( A ) is a G Σ A ,G -weakequivalence of fibration, respectively. We call this the G -global projective levelmodel structure and its weak equivalences G -global level weak equivalences .This model structure is right proper and moreover combinatorial with generatingcofibrations (cid:8) G A (cid:0) (( G × Σ A ) /H × ∂ ∆ n ) + (cid:1) ֒ → G A (cid:0) ( G × Σ A ) /H × ∆ n ) + (cid:1) : n ≥ , H ∈ G Σ A ,G (cid:9) and generating acyclic cofibrations (cid:8) G A (cid:0) (( G × Σ A ) /H × Λ nk ) + (cid:1) ֒ → G A (cid:0) ( G × Σ A ) /H × ∆ n ) + (cid:1) : 0 ≤ k ≤ n, H ∈ G Σ A ,G (cid:9) . Finally, filtered colimits in this model category are homotopical.
We will see in Lemma 3.1.25 below that this model structure is also left proper.
Warning . If G is finite, then being a G -global projective level fibrationor weak equivalence is in some sense orthogonal to being a G -equivariant projectivelevel fibration or weak equivalence, respectively: in level A , the former is a conditionon H -fixed points for H ∈ G Σ A ,G , while the latter one is a condition on H -fixedpoints for H ∈ G G, Σ A . Proof of Proposition 3.1.18.
For a finite set A , we equip ( G × Σ A )-SSet ∗ with the G Σ A ,G -model structure. In order to construct the desired model structureon G -Spectra , to show that it is cofibrantly generated (hence combinatorial) withthe above sets of generating cofibrations and generating acyclic cofibrations, andto verify the above characterization of the cofibrations, it then suffices to verifythat these model categories satisfy the ‘consistency condition’ of Proposition 1.3.5,i.e. that for all finite sets A, B with | A | ≤ | B | and any acyclic cofibration i in the G Σ A ,G -model structure on ( G × Σ A )-SSet ∗ any pushout of Σ ( B, A ) ∧ Σ ( A,A ) i is a G Σ B ,G -weak equivalence. For this it is again enough to show that this is an acycliccofibration in the injective G Σ B ,G -equivariant model structure for every generatingacyclic cofibration, which is obvious.It remains to show that this model structure is right proper, simplicial (i.e. thatthe cotensoring is a right Quillen bifunctor), and that filtered colimits in it arehomotopical. However, these follow easily from the corresponding statement for ( G × Σ A )-SSet with varying A as all the relevant constructions are defined level-wise. (cid:3) The G -global projective level model structure has few cofibrant objects; whilethis will be useful in several arguments, an unfortunate side effect is that manyexamples we care about in practice are not projectively cofibrant: .1. G -GLOBAL HOMOTOPY THEORY OF G -SPECTRA 175 Remark . We claim that the sphere spectrum S is not G -globally pro-jective unless G = 1. Indeed, if A, B are any finite sets, then Σ A acts freely on Σ ( A, B ) outside the base point because it freely permutes the wedge summands.We therefore conclude from the explicit description of G A that each of the standardgenerating cofibrations is of the form X ∧ ( ∂ ∆ n ֒ → ∆ n ) + for some G -spectrum X on which G acts levelwise freely outside the base point.By cell induction one then easily concludes that G acts levelwise freely outsidethe basepoint on any cofibrant object of the G -global projective level model struc-ture. In particular, for G = 1 the only cofibrant spectrum with trivial G -action isthe zero spectrum.To salvage this issue, we will introduce another model structure based on theflat cofibrations: Proposition . There is a (unique) model structure on G -Spectra whosecofibrations are the flat cofibrations and whose weak equivalences are the G -globallevel equivalences. We call this the G -global flat level model structure . This modelstructure is right proper, simplicial, and combinatorial with generating cofibrations (cid:8) G A (cid:0) (( G × Σ A ) /H × ∂ ∆ n ) + (cid:1) ֒ → G A (cid:0) ( G × Σ A ) /H × ∆ n ) + (cid:1) : n ≥ , H ⊂ Σ A × G (cid:9) Moreover, filtered colimits in it are homotopical. Finally, a map f : X → Y is an(acyclic) fibration if and only if f ( A ) : X ( A ) → Y ( A ) is an (acyclic) fibration inthe injective G Σ A ,G -equivariant model structure on ( G × Σ A )-SSet . Proof.
To construct the model structure, it is enough to show that for allfinite sets
A, B with | A | ≤ | B | the functor(3.1.2) Σ ( A, B ) ∧ Σ A – : ( G × Σ A )-SSet ∗ → ( G × Σ B )-SSet ∗ is left Quillen with respect to the injective G Σ A ,G -equivariant model structure onthe source and the injective G Σ B ,G -model structure on the target.For this, we fix a finite set C together with a bijection B ∼ = A ∐ C , whichinduces an injective group homomorphism α : Σ A × Σ C → Σ B . Then [ Hau17 , proofof Proposition 2.24] shows that (3 . .
2) is isomorphic to the composition ( G × Σ A )-SSet ∗ – ∧ S C −−−−→ ( G × Σ A × Σ C )-SSet ∗ ( G × α ) ! −−−−−→ ( G × Σ B )-SSet ∗ and we claim that both of these are left Quillen when we equip the middle termwith the injective G Σ A × Σ C ,G -equivariant model structure. Indeed, it is obvious thatthe first functor preserves injective cofibrations as well as weak equivalences, andfor the second functor it suffices to prove the unbased version, which is a specialcase of Proposition 1.1.42.As in the proof of Proposition 3.1.18 we now get a model structure with thedesired weak equivalences, (acyclic) fibrations, and generating cofibrations. More-over, we conclude as in Proposition 3.1.10 that the cofibrations are precisely theflat cofibrations. Finally, to see that this model structure is right proper, simpli-cial, and that filtered colimits in it are homotopical, one argues as in the projectivesituation above. (cid:3) Again, we will prove later that this model structure is also left proper.
Remark . It is clear that if p is an acyclic fibration in the G -global flatmodel structure, then p ( A ) is in particular a ( G × Σ A )-weak equivalence for all
76 3. STABLE G -GLOBAL HOMOTOPY THEORY finite sets A . Such maps will become useful at several points below and we callthem strong level weak equivalences . The factorization axiom for the above modelstructure then in particular shows that any map of G -spectra factors as a G -globalflat cofibration followed by a strong level equivalence. Warning . While the G -global flat level model structure has the samecofibrations as the G -equivariant flat level model structure (if G is finite), its weakequivalences are once again very different. As a drastic example, if G = 1, then amorphism f of ordinary spectra is a G -weak equivalence if and only if it is levelwisean underlying weak equivalence while it is a G -global weak equivalence if and onlyif each f ( A ) is a Σ A -weak equivalence. For general G , the two notions of weakequivalence are incomparable.In particular, the fibrations of the G -global flat level model structure are verydifferent from the ones of the G -equivariant flat level model structure (the cofreenessconditions are once again ‘orthogonal’ to each other). However we have: Lemma . Let H be a finite group, and let ϕ : H → G be any grouphomomorphism. Then the simplicial adjunction ϕ ! : H -Spectra H -equivariant projective level ⇄ G -Spectra G -global flat level : ϕ ∗ is a Quillen adjunction. In particular, if G is finite, then any fibration or acyclicfibration in the G -global flat level model structure is also a fibration or acyclicfibration, respectively, in the G -equivariant projective level model structure. Proof.
It suffices to prove the first statement. As this is canonically a simpli-cial adjunction, it only remains to prove that ϕ ∗ is right Quillen.A map p of G -spectra is a fibration or acyclic fibration in the G -global flat levelmodel structure if and only if for each finite set A the map p ( A ) is a fibration oracyclic fibration, respectively, in the injective G Σ A ,G -model structure; in particular, p ( A ) is a fibration or acyclic fibration in the A ℓℓ -model structure. But ϕ ∗ : (Σ A × G )-SSet → (Σ A × H )-SSet is right Quillen with respect to the A ℓℓ -model structures, so ( ϕ ∗ p )( A ) = ϕ ∗ ( p ( A ))is a fibration or acyclic fibration in the A ℓℓ -model structure on (Σ A × H )-SSet ,hence in particular in the G G, Σ A -model structure as desired. (cid:3) Lemma . Both of the above model structures on G -Spectra are leftproper. Moreover, pushouts along levelwise injections are already homotopy pushoutsin either model structure. Proof.
Pushouts along levelwise injections preserve G -global level weak equiv-alences because the injective G Σ A ,G -equivariant model structure is left proper forany (finite) set A . The claim follows as before. (cid:3) Stable model structures.
We now turn to the appropriate stable modelstructures on G -Spectra that will then model stable G -global homotopy theory. Definition . A map f : X → Y is called a G -global stable weak equiva-lence if for all finite groups H and all group homomorphisms ϕ : H → G the inducedmap ϕ ∗ f : ϕ ∗ X → ϕ ∗ Y is an H -equivariant stable weak equivalence.For brevity, we will again drop the word ‘stable’ and just use the terms ‘ G -globalweak equivalence’ and ‘ H -equivariant weak equivalence.’ .1. G -GLOBAL HOMOTOPY THEORY OF G -SPECTRA 177 Remark . It is obvious from the 2-out-of-6 property for the H -equivariantweak equivalences that also the G -global weak equivalences satisfy 2-out-of-6 andin particular 2-out-of-3.In general, the G -global weak equivalences are hard to grasp as already the H -equivariant weak equivalences are quite complicated. However, as in the equivariantsetting, there is a notion of π ∗ -isomorphism that is easier to understand and coarseenough to be useful in many practical situations: Definition . Let X be any G -spectrum and let ϕ : H → G be any grouphomomorphism from a finite group H to G . Then we define the ϕ -equivariant(stable) homotopy groups of X as the Z -graded abelian group π ϕ ∗ ( X ) := π H ∗ ( ϕ ∗ X ) . If f : X → Y is a map of G -spectra, then we define π ϕ ∗ ( f ) := π H ∗ ( ϕ ∗ f ). The map f is called a π ∗ -equivalence if π ϕ ∗ ( f ) is an isomorphism for all homomorphisms ϕ : H → G from finite groups H to G . Lemma . Let f : X → Y be a G -global level weak equivalence. Then f isa π ∗ -isomorphism. Proof.
Let H be any finite group and let ϕ : H → G be a group homo-morphism. We will only show that π ϕ ( f ) is an isomorphism, the proof in otherdimensions is analogous, but requires more notation.For this we define s ′ ( U H ) as the poset of all finite faithful H -subsets of U H . Then s ′ ( U H ) is again filtered, and the inclusion s ′ ( U H ) ֒ → s ( U H ) is cofinal (for example,because any H -subset of U H containing our favourite copy of H is faithful). Itfollows that the vertical arrows in the commutative diagramcolim A ∈ s ′ ( U H ) [ S A , ( ϕ ∗ X )( A )] colim A ∈ s ′ ( U H ) [ S A , ( ϕ ∗ Y )( A )]colim A ∈ s ( U H ) [ S A , ( ϕ ∗ X )( A )] colim A ∈ s ( U H ) [ S A , ( ϕ ∗ Y )( A )] incl colim A [ S A , ( ϕ ∗ f )( A )] H ∗ incl π ϕ ( f ) are isomorphisms. On the other hand, if A is a faitful H -set, then ( ϕ ∗ f )( A ) isan H -equivariant weak equivalence. Thus, also the lower horizontal arrow is anisomorphism, and hence so is π ϕ ( f ) as desired. (cid:3) On the other hand, Theorem 3.1.15 implies:
Corollary . Let f be a π ∗ -isomorphism (for example if f is a G -globallevel weak equivalence). Then f is a G -global weak equivalence. (cid:3) In the equivariant setting we have seen a definition of the weak equivalencesin terms of G - Ω -spectra , and a similar characterization exists in the global context,see [ Hau19b , Sections 2.2–2.3]. Here is a G -global generalization of this: Definition . A G -spectrum X is called a G -global Ω -spectrum if thefollowing holds: for any finite group H , any group homomorphism ϕ : H → G , anyfinite faithful H -set A and any finite H -set B the adjoint structure map( ϕ ∗ X )( A ) → R Ω B ( ϕ ∗ X )( A ∐ B )is an H -weak equivalence.
78 3. STABLE G -GLOBAL HOMOTOPY THEORY An equivalent way of formulating the above is the following: for any
H, A, B as above the map X ( A ) → R Ω B X ( A ∐ B ) is a G H,G -weak equivalence. If X is G -globally projectively fibrant (and hence in particular if it is fibrant in the G -global flat level model structure), then ( ϕ ∗ X )( A ∐ B ) is a fibrant H -simplicial setand X ( A ∐ B ) is fibrant in the G H,G -model structure on ( G × H )-simplicial sets(both of these use that A ∐ B is faithful). Thus, we can in this case just work withnon-derived loop space in either of the above equivalent definitions. Proposition . A map f : X → Y is a G -global weak equivalence if andonly if for every G -global Ω -spectrum T the induced map [ f, T ] : [ Y, T ] → [ X, T ] isbijective, where [ , ] denotes the hom sets in the homotopy category with respect tothe G -global level weak equivalences. The proof will be given later once we have constructed the relevant modelstructures.
Lemma . Let H be a finite group and ϕ : H → G any homomorphism.Then the simplicial adjunction ϕ ∗ : G -Spectra G -global flat level ⇄ H -spectra H -equivariant flat : ϕ ∗ is a Quillen adjunction with homotopical left adjoint. Moreover, the essential imageof R ϕ ∗ is contained in the class of G - Ω -spectra. Proof.
It is obvious that ϕ ∗ preserves flat cofibrations and it is actually ho-motopical by Lemma 3.1.30, hence in particular left Quillen. To finish the proof itsuffices now that for any fibrant spectrum X in the H -equivariant flat model struc-ture the G -spectrum ϕ ∗ X is a G -Ω-spectrum, which by the Quillen adjunction justestablished amounts to saying that the adjunct structure map(3.1.3) ( ϕ ∗ X )( A ) → Ω B ( ϕ ∗ X )( A ∐ B )is a G K,G -weak equivalence for any finite group K to G , any finite faithful K -set A , and any finite K -set B . We will show that it is in fact even a ( K × G )-weakequivalence.Indeed, by [ Hau19b , proof of Proposition 2.14] the adjoint structure map(3.1.4) X ( A ) → Ω B X ( A ∐ B )is actually a ( K × H )-weak equivalence. Moreover, both sides are easily seen to befibrant in the A ℓℓ -model structure on ( K × H )-SSet . As ϕ ∗ : ( K × G )-SSet → ( K × H )-SSet is left Quillen for the A ℓℓ -model structures on both sides, its rightadjoint ϕ ∗ is right Quillen, so it sends (3 . .
4) to a ( K × G )-weak equivalence by KenBrown’s Lemma. This image agrees with (3 . .
3) up to conjugation by isomorphisms(as ϕ ∗ preserves cotensors), which completes the proof of the lemma. (cid:3) Lemma . Let f : X → Y be a map of G - Ω -spectra. Then f is a G -globallevel weak equivalence if and only if f is a G -global weak equivalence. Proof.
The implication ‘ ⇒ ’ holds by Lemma 3.1.30 even without any assump-tions on X and Y .For the implication ‘ ⇐ ’, we may assume without loss of generality by 2-out-of-3that X and Y are fibrant in the G -global flat level model structure. Now let H be .1. G -GLOBAL HOMOTOPY THEORY OF G -SPECTRA 179 any finite group, let ϕ : H → G be a group homomorphism, and let A be a finitefaithful H -set. We consider the diagram ϕ ∗ X Ω A sh A ϕ ∗ Xϕ ∗ Y Ω A sh A ϕ ∗ Y. λ X ϕ ∗ f Ω A sh A ϕ ∗ fλ Y with λ X given in degree B by the adjunct structure map ϕ ∗ X ( B ) → Ω A ( ϕ ∗ X )( A ∐ B ), and analogously for Y . Thus, λ X and λ Y are H -global level weak equivalencesby assumption on X and Y , and hence in particular H -stable weak equivalencesby Lemma 3.1.30. Moreover, ϕ ∗ f is an H -stable weak equivalence by definition,so that Ω A sh A ϕ ∗ f is an H -stable weak equivalence by 2-out-of-3. As both itssource and target are H -Ω-spectra by definition, Ω A sh A ϕ ∗ f is an H -equivariantlevel weak equivalence [ Hau17 , Remark 2.36]. In particular, if B is any finite H -set, then (Ω A sh A ϕ ∗ f )( B ) is a H -equivariant weak equivalence. But this fits into acommutative diagram( ϕ ∗ X )( B ) (Ω A sh A ϕ ∗ X )( B )( ϕ ∗ Y )( B ) (Ω A sh A ϕ ∗ X )( B ) , λ X ( B )( ϕ ∗ f )( B ) (Ω A sh A ϕ ∗ f )( B ) λ Y ( B ) and if B is actually faithful , then also the horizontal arrows are H -weak equivalencesby the above. It follows by 2-out-of-3 that in this case also the left hand verticalarrow is an H -equivariant weak equivalence as desired. (cid:3) Construction . Let H be a finite group, let A be a finite H -set, and let ϕ : H → G be any homomorphism. Then adjointness yields a natural isomorphism(–) ϕ ◦ ev A ∼ = G A (( G × Σ A ) /H + ).If B is another finite H -set, then G A ∐ B (cid:0) ( S B ∧ ( G × Σ A ∐ B ) + ) /H (cid:1) similarlycorepresents T (Ω B T ( A ∐ B )) ϕ . Thus the Yoneda Lemma yields a map λ BA,ϕ : G A ∐ B (cid:0) ( S B ∧ ( G × Σ A ∐ B ) + ) /H (cid:1) → G A (( G × Σ A ) + /H such that the restriction maps G ( λ BA,ϕ , T ) is conjugate to the ϕ -fixed points of theadjunct structure map T ( A ) → Ω B T ( A ∐ B ) for any G -spectrum T .We can now finally prove: Theorem . There is a unique model structure on G -Spectra whosecofibrations are the flat cofibrations and whose weak equivalences are the G -globalweak equivalences. We call this the G -global flat model structure . It is left proper,combinatorial, simplicial, and filtered colimits in it are homotopical. Moreover, a G -spectrum is fibrant in it if and only if it is fibrant in the G -global flat level modelstructure and moreover a G -global Ω -spectrum. Proof.
It is clear that the sources and targets of the maps λ BA,ϕ from theprevious construction are cofibrant in the projective G -global level model structure,and hence in particular in the flat G -global level model structure. We may thereforeapply Theorem A.2.4 to obtain a left proper, simplicial, and combinatorial modelstructure on G -Spectra with the same cofibrations as the flat G -global level model
80 3. STABLE G -GLOBAL HOMOTOPY THEORY structure and whose fibrant objects are precisely those G -spectra X that are fibrantin the G -global flat level model structure and for which maps G ( λ BA,ϕ ) is a weakhomotopy equivalence for all
A, B, ϕ as above. By construction, this is equivalentto X being a G -global Ω-spectrum.Let us call the weak equivalences of this model structure G -global Ω -weak equiv-alences for the time being. Lemma A.2.6 shows that they are stable under filteredcolimits, and it only remains to show that they agree with the G -global weak equiv-alences, which amounts to saying that(3.1.5) ( ϕ ∗ ) ϕ : H → G : G -Spectra G -global Ω-w.e. → Y ϕ : H → G H -Spectra H -equiv. w.e. preserves and reflects weak equivalences, where the product runs over a system ofrepresentatives of finite groups H , and all homomorphisms ϕ : H → G .Indeed, let us equip the left hand side with the above model structure and theright hand side with the product of the H -equivariant flat model structures; weclaim that (3 . .
5) is left Quillen, for which it suffices to show that ( ϕ ∗ ) ϕ preservescofibrations and that its right adjoint (which sends a family ( X ϕ ) ϕ of H -spectra tothe product Q ϕ ϕ ∗ X ϕ ) preserves fibrant objects. But as fibrant objects are stableunder small products, this is immediate from Lemma 3.1.33. As ( ϕ ∗ ) ϕ moreoversends G -global level weak equivalences to weak equivalences by the same lemma,we conclude that it is homotopical in G -global Ω-weak equivalences.It therefore only remains to show that (3 . .
5) also reflects weak equvialences.But since we already know that it is homotopical, it suffices to show this for mapsbetween fibrant objects, in which case the claim follows from 3.1.34. (cid:3)
Proof of Proposition 3.1.32.
This is immediate from the description ofthe weak equivalences of the above model structure provided by Theorem A.2.4. (cid:3)
Corollary . The G -global weak equivalences are stable under all smallcoproducts. Proof.
This is clear for the G -global level weak equivalences, so the coprod-ucts in G -Spectra also define coproducts in the localization with respect to the G -global level weak equivalences. The claim now follows immediately from theprevious characterization. (cid:3) Theorem . There is a unique model structure on G -Spectra whosecofibrations are the G -global projective cofibrations and whose weak equivalencesare the G -global weak equivalences. We call this the G -global projective modelstructure . It is left proper, combinatorial, simplicial, and filtered colimits in it arehomotopical. Moreover, a G -spectrum is fibrant in it if and only if it is fibrant inthe G -global projective level model structure and moreover a G -global Ω -spectrum. Proof.
Analogously to Theorem 3.1.36, one constructs a model structure withthe desired cofibrations and fibrant objects, and shows that it is left proper, com-binatorial, simplicial, and that filtered colimits in it are homotopical. The weakequivalences of this model structure can then again be detected by mapping into G -global Ω-spectra, so Proposition 3.1.32 shows that they agree with the G -globalweak equivalences. (cid:3) Remark . If we fix a finite group G , then we can equip G -Spectra witheither the G -global flat model structure or the G -equivariant flat model structure. .1. G -GLOBAL HOMOTOPY THEORY OF G -SPECTRA 181 Both of these have the same cofibrations, but the G -global weak equivalences arestrictly finer than the G -equivariant ones even for G = 1.As a model structure is characterized by its cofibrations together with its fibrantobjects, it follows that being fibrant in the G -global flat model structure is strictly weaker than being fibrant in the G -equivariant flat model structure. However, wewill show later in Proposition 3.3.1 that a fibrant object in the G -global flat modelstructure is at least still a G -Ω-spectrum. Lemma . Any pushout along a levelwise cofibration is a homotopy pushoutin both the G -global projective model structure as well as the G -global flat modelstructure. In particular, pushouts of G -global weak equivalences along levelwisecofibrations are again G -global weak equivalences. Proof.
This follows from Lemma 3.1.25 as in the proof of Theorem 1.2.41. (cid:3)
Corollary . There is a unique model structure on G -Spectra whoseweak equivalences are the G -global weak equivalences and whose cofibrations are thelevelwise cofibrations. We call this the injective G -global model structure . It iscombinatorial, simplicial, left proper, and filtered colimits in it are homotopical. Proof.
The only non-trivial statement is that this model structure is simpli-cial. To this end, it suffices as before that for each K ∈ SSet ∗ the functor K ∧ – ishomotopical and that for each G -spectrum X so is – ∧ X . But indeed, the secondstatement is obvious, and for the first we observe that K ∧ – preserves G -globalprojective acyclic cofibrations (by Theorem 3.1.38) as well as G -global level weakequivalences (by an easy calculation). The claim follows. (cid:3) Proposition . Let (3.1.6)
P XY Z g pf be a pullback diagram in G -Spectra such that p is a G -global projective level fibra-tion and f is a G -global weak equivalence. Then g is a G -global weak equivalence,too. In particular, the G -global projective model structure, the G -global flat modelstructure, and the G -global injective model structure all are right proper (henceproper). Proof.
It suffices to prove the first statement. For this we employ the factor-ization axiom of the G -global flat level model structure to write f as a composition Y i −→ H s −→ Z, where i is a G -global level weak equivalence and s is a fibration in the G -global flatlevel model structure. We observe that s is moreover a G -global weak equivalenceby Lemma 3.1.30 together with 2-out-of-3. The diagram (3 . .
6) now factors as
P I XY H Z j tq pi s
82 3. STABLE G -GLOBAL HOMOTOPY THEORY where both squares are pullbacks. In particular, q is again a G -global projectivelevel fibration and hence j is a G -global level weak equivalence by right proper-ness of the G -global projective level model structure. By another application ofLemma 3.1.30 and 2-out-of-3, it is therefore enough to show that t is a G -globalweak equivalence.For this let ϕ : H → G be any homomorphism from a finite group H to G ; then ϕ ∗ I ϕ ∗ Xϕ ∗ H ϕ ∗ Z ϕ ∗ tϕ ∗ q ϕ ∗ pϕ ∗ s is a pullback and ϕ ∗ s is an H -equivariant projective level fibration (by Lemma 3.1.24)as well as an H -equivariant stable weak equivalence (by the above). It there-fore follows from [ Hau17 , discussion after Proposition 4.2] that also ϕ ∗ t is an H -equivariant stable weak equivalence. Letting ϕ vary we see therefore see that t is a G -global weak equivalence, finishing the proof. (cid:3) Finally we observe:
Proposition . The G -global projective model structure, the G -global flatmodel structure, and the G -global injective model structure all are stable. Proof.
Each of these are simplicial model categories, so this amounts to sayingthat the suspension/loop adjunction Σ := S ∧ – ⊣ maps( S , –) := Ω (which isalways a Quillen adjunction) is a Quillen equivalence for each of these. As the leftadjoint is homotopical and all of these have the same weak equivalences, it sufficesto show:(1) For every G -spectrum X and some (hence any) fibrant replacement Σ X → Y in the G -global flat model structure, the composition X → ΩΣ X → Ω Y is a G -global weak equivalence.(2) For every G -spectrum Y that is fibrant in the G -global flat model struc-ture, the counit ΣΩ X → X is a G -global weak equivalence.We will prove the first statement, the argument for the second one being analogous.If f : Σ X → Y is any fibrant replacement in the G -global flat model structure,then ϕ ∗ ( f ) is an H -equivariant weak equivalence and ϕ ∗ ( Y ) is fibrant in the H -equivariant projective level model structure for every homomorphism ϕ : H → G .By Proposition 3.1.17, Σ is also homotopical with respect to H -equivariant weakequivalences, and Ω preserves H -equivariant weak equivalences between objectsthat are fibrant in the H -equivariant projective level model structure; we thereforeconclude that ϕ ∗ ( f η ) is a model for the derived unit of the suspension loop ad-junction on H -Spectra H -equiv. proj. . The claim follows by another application ofProposition 3.1.17. (cid:3) We will now discuss how the above models relate toeach other when the group G varies. Lemma . For any homomorphism α : H → G , the simplicial adjunction α ! : H -Spectra H -global projective ⇄ G -Spectra G -global projective : α ∗ is a Quillen adjunction, and likewise for the corresponding level model structures. .1. G -GLOBAL HOMOTOPY THEORY OF G -SPECTRA 183 Proof.
It is clear that α ∗ is right Quillen, proving the second statement. Forthe first statement it then suffices that α ∗ preserves fibrant objects, which by theabove together with the characterization of fibrant objects amounts to saying that α ∗ sends G -globally projectively fibrant G -global Ω-spectra to H -global Ω-spectra.This is obvious from the definitions. (cid:3) Lemma . For any homomorphism α : H → G , the simplicial adjunction α ∗ : G -Spectra G -global flat ⇄ H -Spectra H -global flat : α ∗ . is a Quillen adjunction. (cid:3) Unfortunately, α ∗ is not in general right Quillen with respect to the correspond-ing flat model structures: Example . We let X be any globally fibrant spectrum whose underlyingnon-equivariant stable homotopy type agrees with the Eilenberg-Mac Lane spec-trum H Z /
2. As globally fibrant spectra are in particular non-equivariant Ω-spectra(the trivial group acts faithfully on the empty set) and levelwise Kan complexes, weconclude that X ( ∗ ) represents the first singular cohomology group with coefficientsin Z /
2, i.e. π maps(– , X ( ∗ )) ∼ = H (– , Z / α : Z / → α ∗ X is not even fibrant in the Z / level model structure: namely, this wouldin particular mean thatmaps Z / ( E ( Z / , ( α ∗ X )( ∗ )) ≃ maps Z / ( ∗ , ( α ∗ X )( ∗ ))as E ( Z / → ∗ is a G Σ ∗ , Z / -equivariant (i.e. underlying) weak equivalence. Thus,maps( E ( Z / / ( Z / , X ( ∗ )) ∼ = maps Z / ( E G , Z / , ( α ∗ X )( ∗ )) ≃ maps Z / ( ∗ , ( α ∗ X )( ∗ )) = maps( ∗ , X ( ∗ )) . However, E ( Z /
2) is a contractible space with free Z / Z / K ( Z / ,
1) and hence equivalent to R P ∞ . Thus taking π of the above wecould conclude that H ( R P ∞ , Z / ∼ = H ( ∗ , Z / Proposition . Let α : H → G be an injective group homomorphism.Then the simplicial adjunction α ! : H -Spectra flat H -global ⇄ G -Spectra flat G -global : α ∗ is a Quillen adjunction and likewise for the corresponding level model structures. Proof.
For the second statement it suffices to observe that α ! : (Σ A × H )-SSet inj. G Σ A,H -equiv. ⇄ (Σ A × G )-SSet inj. G Σ A,G -equiv. : α ∗ , is a Quillen adjunction for every finite set A by Proposition 1.1.42. With thisestablished, it suffices for the first statement to prove that α ∗ sends fibrant objectsto H -global Ω-spectra. This holds in fact for any group homomorphism α byLemma 3.1.44. (cid:3) The same example as above shows that α ∗ is not right Quillen with respect tothe corresponding injective model structures. However, we still have:
84 3. STABLE G -GLOBAL HOMOTOPY THEORY Proposition . Let α : H → G be an injective group homomorphism.Then the simplicial adjunction α ! : H -Spectra injective H -global ⇄ G -Spectra injective G -global : α ∗ is a Quillen adjunction. Proof.
It is obvious that α ! preserves levelwise injections. To prove that it ishomotopical, we use the factorization axiom in the flat H -global model structure tofactor any given H -global weak equivalence f as an acyclic H -global flat cofibration i followed by a strong level weak equivalence p . By Proposition 3.1.47, α ! ( i ) is a G -global weak equivalence, and so is α ! ( p ) for obvious reasons. (cid:3) The functor α ! preserves injective cofibrations in full generality, so the failureof α ∗ to be right Quillen with respect to the injective model structures can bereinterpreted as a failure of α ! to be homotopical. However, similarly to the unstablesituation we have: Proposition . Let α : H → G be any homomorphism, and let f : X → Y be an H -global weak equivalence such that ker( α ) acts levelwise freely on X and Y outside the base point. Then α ! f is a G -global weak equivalence. Proof.
Lemma 3.1.44 in particular implies that α ! preserves weak equivalencesbetween H -globally projectively cofibrant objects. Choosing functorial factoriza-tions in the H -global projective model structure, it therefore suffices to show: if p : X ′ → X is any H -global level weak equivalence such that X ′ is cofibrant in the H -global projective model structure, then α ! ( p ) is a G -global weak equivalence.Indeed, if A is any finite set, then p ( A ) is a G Σ A ,H -weak equivalence, and ker α acts freely outside the basepoint on X ( A ) by assumption. On the other hand, asseen in Remark 3.1.20, all of H acts freely on X ′ ( A ), so the claim is an instance ofRemark 2.2.20. (cid:3) The stability ofany of the G -global model structures in particular implies that the G -global stablehomotopy category is additive, i.e. finite coproducts and products agree in it. Thefollowing lemma provides an underived version of this: Lemma . Let T be a finite set and let ( X t ) t ∈ T be any family of G -spectra.Then the canonical map W t ∈ T X t → Q t ∈ T X t is a G -global weak equivalence (andin fact a π ∗ -isomorphism). Proof.
Let ϕ : H → G be a group homomorphism from a finite group H to G . Then restricting the canonical map along ϕ agrees with the canonicalmap W t ∈ T ϕ ∗ ( X t ) → Q t ∈ T ϕ ∗ ( X t ) (on the nose, if we use the usual construc-tion of colimits, up to conjugation by isomorphism in general). As the latter is a π H ∗ -isomorphism by [ Hau17 , Proposition 3.6-(3)] and hence in particular an H -equivariant weak equivalence, the claim follows. (cid:3) Together with Corollary 3.1.37 we in particular conclude:
Corollary . Finite products preserve G -global weak equivalences. (cid:3) Remark . In the same way, one deduces from the equivariant comparisoncited above together with [
Hau17 , Proposition 4.2-(1)] that finite products preserve H -equivariant weak equivalences for any finite group G . .1. G -GLOBAL HOMOTOPY THEORY OF G -SPECTRA 185 If X is any G -spectrum, then we can in particular apply Lemma 3.1.50 to thefamily constant at X , proving that(3.1.7) _ T X → Y T X is a G -global weak equivalence. As for G -global Γ-spaces, we want to strengthenthis to a comparison taking the Σ T -symmetries into account: Proposition . The canonical map (3 . . is a ( G × Σ T ) -global weakequivalence (in fact, even a π ∗ -isomorphism) with respect to the Σ T -actions per-muting the summands or factors, respectively. While this is not the easiest way to prove the proposition, we will deduce itfrom a suitable version of the Wirthm¨uller isomorphism again. More precisely, recall(Construction 2.2.66) the construction of the natural Wirthm¨uller map γ : α ! Y → α ∗ Y for any injective homomorphism α : H → G and any pointed H -set Y . Ap-plying this levelwise in the simplicial and spectral direction, we then obtain aWirthm¨uller map γ : α ! X → α ∗ X for any H -spectrum. We are then going toprove more generally: Theorem . Let α : H → G be an injectivehomomorphism such that ( G : im α ) < ∞ , and let X be any H -spectrum. Then theWirthm¨uller map γ : α ! X → α ∗ X is a G -global weak equivalence (and in fact evena π ∗ -isomorphism). The proof of the theorem needs some preparations, but before we come to this,let us use it to prove the proposition:
Proof of Proposition 3.1.53.
This follows from Theorem 3.1.54 by thesame argument as in the proof of Corollary 2.3.6. (cid:3)
We will prove the Wirthm¨uller isomorphis by reduction to the correspondingequivariant statement; a similar argument appears in [
DHL + , proof of Theo-rem 2.1.10]. This will need the following classical pointset level manifestation ofthe Mackey double coset formula whose proof we leave to the interested reader, alsosee [ Hau17 , discussion after Definition 1.5] where this is stated without proof forfinite G . Construction . Let
K, H ⊂ G be any two subgroups and let A be apointed H -set. We define a K -equivariant map α : _ [ g ] ∈ K \ G/H K + ∧ K ∩ gHg − c ∗ g ( A | g − Kg ∩ H ) → ( G + ∧ H A ) | K , where the wedge runs over a fixed choice of double coset representatives, and c g denotes the conjugation homomorphism K ∩ gHg − → g − Kg ∩ H, k g − kg , asfollows: on the summand corresponding to g ∈ G , the map is given in each degreeby [ k, a ] [ kg, a ].Analogously, we define β : maps H ( G, A ) (cid:12)(cid:12) K → Y [ g ] ∈ K \ G/H maps K ∩ gHg − (cid:0) K, c ∗ g ( A | g − Kg ∩ H ) (cid:1) as the map given on the factor corresponding to g ∈ G by evaluation at g .
86 3. STABLE G -GLOBAL HOMOTOPY THEORY We omit the easy verification that these are indeed well-defined, K -equivariant,and moreover isomorphisms. Again, we get spectral versions (denoted by the samesymbols) by applying these levelwise. Construction . Let ϕ : K → G be a surjective group homomorphismand let H ⊂ G be any subgroup. Then we define for any pointed H -set A a map δ : K + ∧ ϕ − H ( ϕ | ∗ ϕ − ( H ) A ) → ϕ ∗ ( G + ∧ H A )via δ [ k, a ] = [ ϕ ( k ) , a ]. Moreover, we write ǫ : ϕ ∗ (cid:0) maps H ( G, X ) (cid:1) → maps ϕ − ( H ) (cid:0) K, ( ϕ | ϕ − ( H ) ) ∗ X (cid:1) for the restriction along ϕ . We again omit the easy verification, that these arewell-defined, K -equivariant, and natural isomorphisms. As before, we get naturalisomorphisms of K -spectra, denoted by the same symbols. Proof of Theorem 3.1.54.
Let ϕ : H → G be any group homomorphism;we have to show that ϕ ∗ ( γ ) is an H -equivariant weak equivalence.For this we factor ϕ as the composition of the surjection ¯ ϕ : H → ϕ ( H ) =: K followed by the inclusion i : K ֒ → H . One then immediately checks that the diagram W [ g ] ∈ K \ G/H K + ∧ K ∩ gHg − c ∗ g ( X | g − Kg ∩ H ) ( G + ∧ H X ) | K maps H ( G, X ) | K Q [ g ] ∈ K \ G/H K + ∧ K ∩ gHg − c ∗ g ( X | g − Kg ∩ H ) Q [ g ] ∈ K \ G/H maps K ∩ gHg − (cid:0) K, c ∗ g ( X | g − Kg ∩ H ) (cid:1) α canonical γβ Q γ in K -Spectra commutes. As seen above, α , β are K -equivariant isomorphisms;moreover, K \ G/H is finite by assumption on H , and hence the left hand verticalarrow in the above diagram is a K -global weak equivalence (and in fact a π ∗ -isomorphism) by Lemma 3.1.50. It is therefore enough to show that the lowerhorizontal map becomes an H -equivariant weak equivalences after restricting along¯ ϕ , for which it is enough (Remark 3.1.52) to show this for each individual factor.Up to renaming, we are therefore reduced to the case that ϕ : K → G is surjective.But in this case, the diagram K + ∧ ϕ − ( H ) ( ϕ | ϕ − ( H ) ) ∗ X maps ϕ − ( H ) (cid:0) K, ( ϕ | ϕ − ( H ) ) ∗ X (cid:1) ϕ ∗ ( G + ∧ H X ) ϕ ∗ (cid:0) maps H ( G, X ) (cid:1) γδ ϕ ∗ ( γ ) ǫ commutes by direct inspection. The vertical arrows were seen to be K -equivariantisomorphisms above, and the top horizontal arrow is a K -equivariant weak equiv-alence (and in fact a π ∗ -isomorphism) by the usual Wirthm¨uller isomorphism, seee.g. [ Hau17 , Proposition 3.7] for a proof attributed to Schwede. The claim there-fore follows by 2-out-of-3. (cid:3) .1. G -GLOBAL HOMOTOPY THEORY OF G -SPECTRA 187 The ordinary smash product of spectra (whichwe recall to mean symmetric spectra!) gives rise to a smash product on G -spectraby functoriality, and analogously for its right adjoint, the function spectrum con-struction F . Explicitly, F ( X, Y )( A ) = maps( X, sh A Y ), where here—and in thediscussion below—we agree on the convention that maps denotes the simplicial setof not necessarily G -equivariant maps , with the conjugation action.We will now discuss some model categorical properties of these two functors: Proposition . (1) If X is any flat G -spectrum, then X ∧ – pre-serves G -global weak equivalences. (2) If X is any spectrum, then X ∧ – preserves G -global weak equivalencesbetween flat G -spectra. Proof.
For the first statement, we let f : Y → Y ′ be any G -global weakequivalence. If ϕ : H → G is any group homomorphism, then we have to showthat ϕ ∗ ( X ∧ f ) is an H -equivariant weak equivalence. But this literally agrees with ϕ ∗ ( X ) ∧ ϕ ∗ ( f ), and as ϕ ∗ ( X ) is flat and ϕ ∗ ( f ) is an H -weak equivalence, the claimfollows from the usual equivariant Flatness Theorem [ Hau17 , Proposition 6.2-(i)].The second statement follows similarly from [
Hau17 , Proposition 6.2-(ii)]. (cid:3)
Proposition . (1) The smash product is a left Quillen bifunctorwith respect to the G -global flat model structures everywhere. (2) The function spectrum is a right Quillen bifunctor with respect to the G -global flat model structures everywhere. Proof.
It suffices to prove the first statement, for which we want to verify thepushout-product axiom. However, as in the previous proof one readily reduces thisto the corresponding equivariant statement, which appears for example as [
Hau17 ,Proposition 6.1]. (cid:3)
Next, we want to establish the analogue of the above result with respect to the G -global projective model structures. In fact, we will prove a stronger ‘mixed’ ver-sion of this, that (in the guises of Corollary 3.1.61 and Theorem 3.2.18) will becomecrucial later in the proof of the G -global Delooping Theorem (Theorem 3.4.22). Proposition . (1) The smash product is a left Quillen bifunctor – ∧ – : G -Spectra G -global flat × G -Spectra G -global proj. → G -Spectra G -global proj. . (2) The function spectrum is a right Quillen bifunctor F : G -Spectra op G -global flat × G -Spectra G -global proj. → G -Spectra G -global proj. . The proof will rely on the following observation:
Lemma . Let A be a finite faithful G -set (in particular G is finite). Then sh A : G -Spectra G -global projective → G -Spectra G -global flat preserves acyclic fibrations. Proof.
Let p : X → Y be an acyclic fibration in the G -global projective modelstructure. We have to show that (sh A p )( B ) H is an acyclic Kan fibration for everyfinite set B and any subgroup H ⊂ G × Σ B .
88 3. STABLE G -GLOBAL HOMOTOPY THEORY But by definition the G × Σ B -action on (sh A p )( B ) = p ( A ∐ B ) is given byrestricting the G × Σ A ∐ B -action on the right hand side along the homomorphism ϕ : G × Σ B G × Σ A × Σ B G × Σ A ∐ B ( g, σ ) ( g, ρ ( g ) , σ )(where ρ : G → Σ A classifies the G -action on A ), so that (sh A p )( B ) H = p ( A ∐ B ) ϕ ( H ) . It therefore suffices that ϕ ( H ) ∈ G Σ A ∐ B ,G . But indeed, if ϕ ( h, σ ) = ( g, σ = 1 and ρ ( h ) = 1 by definition of ϕ . As A is faithful, ρ is injective, so that h = 1 and hence also g = 1. This finishes the proof. (cid:3) Proof of Proposition 3.1.59.
Let i : X → X ′ be a flat cofibration and p : Y → Y ′ be an acyclic fibration of the G -global projective model structure.We will first show that that the ‘pullback corner map’( p ∗ , i ∗ ) : F ( X ′ , Y ) → F ( X ′ , Y ′ ) × F ( X,Y ′ ) F ( X, Y )is a G -global acyclic projective fibration, i.e. if ϕ : H → G is any group homo-morphism and A a finite faithful H -set, then (cid:0) ϕ ∗ ( p ∗ , i ∗ )( A ) (cid:1) H is an acyclic Kanfibration. Indeed, this map agrees up to conjugation by isomorphisms with((sh A ϕ ∗ ( p )) ∗ , ϕ ∗ ( i ) ∗ ) : maps H ( ϕ ∗ X ′ , sh A ϕ ∗ Y ) → maps H ( ϕ ∗ X ′ , sh A ϕ ∗ Y ′ ) × maps H ( ϕ ∗ X, sh A ϕ ∗ Y ′ ) maps H ( ϕ ∗ X, sh A ϕ ∗ Y ) . By the previous lemma together with Lemma 3.1.44, sh A ϕ ∗ ( p ) is an acyclic fibrationin the G -global flat model structure, and moreover ϕ ∗ ( i ) is a flat cofibration bydefinition. The claim follows because the H -global flat model structure is simplicial.By adjointness we may therefore conclude that the pushout product j (cid:3) k : ( A ∧ B ′ ) ∐ ( A ∧ B ) ( A ′ ∧ B ) → A ′ ∧ B ′ of any G -global flat cofibration j : A → A ′ with any G -global projective cofibration k : B → B ′ is a G -global projective cofibration. On the other hand, if at least oneof j or k is acyclic, then Proposition 3.1.58 shows that j (cid:3) k is a G -global acyclicflat cofibration (as any G -global projective cofibration is also a flat cofibration),hence in particular a G -global weak equivalence. This proves the first part of theproposition, and the second one then follows by the usual adjointness argument. (cid:3) If X is an H -spectrum and Y is a G -spectrum, then F ( X, Y ) carries an ( H × G )-action, yielding a functor F : H -Spectra op × G -Spectra → ( H × G )-Spectra .As an application of the above proposition we can now prove: Corollary . (1) If X is any flat H -spectrum, then (3.1.8) F ( X, –) : G -Spectra G -gl. proj. → ( H × G )-Spectra ( H × G )-gl. proj. is right Quillen. (2) If Y is fibrant in the G -global projective model structure, then F (– , Y ) : H -Spectra op H -global flat → ( H × G )-Spectra ( H × G )-global projective is right Quillen. Proof.
We will prove the first statement, the proof of the second being similar. .2. CONNECTIONS TO UNSTABLE G -GLOBAL HOMOTOPY THEORY 189 For this we let pr : H × G → H and pr : H × G → G denote the projections.Then (3 . .
8) agrees with the composition G -Spectra G -global proj. pr ∗ −−→ ( H × G )-Spectra ( H × G )-global proj. F (pr ∗ X, –) −−−−−−−→ ( H × G )-Spectra ( H × G )-global proj. . By Lemma 3.1.44 the first functor is right Quillen, and so is the second one byProposition 3.1.59 as pr ∗ X is flat. (cid:3) Remark . By another adjointness argument, Proposition 3.1.59 impliesthat F is also a right Quillen bifunctor with respect to the G -global projective modelstructures on the source and the G -global flat model structure in the target. Whilethis may sound somewhat odd, we emphasize again that the G -global projectivemodel structure has ‘few’ cofibrant objects for G = 1. In particular, S is notcofibrant, so this does not apply to F ( S , –) ∼ = id. G -global homotopy theory We can now connect the above stable models to the unstable ones discussed inChapter 1, and in particular to G - I -simplicial sets and G - I -simplicial sets. We begin by recalling the suspension/loop ad-junction between I -SSet ∗ and Spectra , see e.g. [
SS12 , 3.17]:
Construction . if X is any I -simplicial set, then we define a symmetricspectrum Σ • X via (Σ • X )( A ) = S A ∧ X ( A ) together with the evident structuremaps; Σ • becomes a simplicially enriched functor in the obvious way. On the otherhand, if Y is a symmetric spectrum, then we define Ω • Y via (Ω • Y )( A ) = Ω A Y ( A )together with the evident structure maps. Again, this becomes a functor in theobvious way, and we omit the easy verification that Σ • is left adjoint to Ω • .By functoriality, this gives rise to an adjunction G - I -SSet ∗ ⇄ G -Spectra that we denote by Σ • ⊣ Ω • again. Proposition . The simplicial adjunction (3.2.1) Σ • : ( G - I -SSet ∗ ) G -global ⇄ G -Spectra G -global projective : Ω • is a Quillen adjunction, and Σ • is fully homotopical. Proof.
One immediately checks from the definitions that S A ∧ – : ( G × Σ A )-SSet ∗ ⇄ ( G × Σ A )-SSet ∗ : Ω A (where Σ A acts on S A via its tautological A -action) is a Quillen adjunction withrespect to the G Σ A ,G -equivariant model structure on both sides, proving that (3 . . G -global projective level model structure on the target, hence also with respectto the G -global projective model structure on the target. To prove that also (3 . . • sends fibrant objects to static G - I -simplicial sets, which is immediate from the definitions.To show that Σ • is homotopical, it is now enough to observe that any weakequivalence in G - I -SSet ∗ factors as an acyclic cofibration followed by a strict levelweak equivalence. The former are preserved by the above, and clearly strict levelweak equivalences are even sent to G -global level weak equivalences. (cid:3)
90 3. STABLE G -GLOBAL HOMOTOPY THEORY Corollary . The simplicial adjunction (3 . . is also a Quillen adjunc-tion with respect to the G -global injective model structures. Proof.
It is obvious that Σ • preserves injective cofibrations, and by the pre-vious proposition it also preserves weak equivalences. (cid:3) Remark . Assume G is finite, let X be any G -spectrum, and let U be ourfixed complete G -set universe. Then we have preferred maps π G (cid:0) (Ω • X )( U ) (cid:1) ∼ = π G (cid:0) colim A ∈ s ( U ) Ω A X ( A ) (cid:1) ∼ = colim A ∈ s ( U ) π G Ω X ( A ) = colim A ∈ s ( U ) π maps G ∗ ( S A , X ( A )) → colim A ∈ s ( U ) [ S A , X ( A )] G ∗ = π G ( X )(where the first isomorphism comes from cofinality, the second one uses that π G preserves filtered colimits, and the final map is given by geometric realization),yielding a natural transformation π G ◦ Ω • ⇒ π G . If X is fibrant in the G -globalprojective level model structure, then X ( A ) is fibrant in the G -equivariant modelstructure for every finite faithful G -set A , so that the final map is an isomorphismas a colimit of (eventual) isomorphisms, and hence so is the above composition.We now turn to a variant for G - I -simplicial sets: Corollary . The simplicial adjunction Σ • := Σ • ◦ forget: ( G - I -SSet ∗ ) G -global ⇄ G -Spectra G -global : maps I ( I , –) ◦ Ω • is a Quillen adjunction, and likewise for the injective G -global model structures. Proof.
This is immediate from the above as the forgetful functor G - I -SSet → G - I -SSet is left Quillen for the G -global model structures as well as for the injec-tive G -global ones (Corollary 1.4.30 and Theorem 1.3.44, respectively). (cid:3) Remark . Composing with the adjunction(–) + : G - I -SSet ⇄ G - I -SSet + : forget(or its I -version) we also get unpointed versions of all of the above adjunctions.We denote the right adjoints as before and write Σ • + for the left adjoints. Remark . By Corollary 1.4.30, the counit forget maps I ( I , Y ) → Y is a G -global weak equivalence for any G - I -simplicial set Y fibrant in the G -global modelstructure. By the previous remark, this in particular applies to Y = Ω • X for any G -spectrum X fibrant in the G -global projective model structure. Composing withthe transformation from Remark 3.2.4 we therefore get a natural transformation π G (cid:0) maps I ( I , Ω • (–))( U ) (cid:1) ⇒ π G , and this is an isomorphism on any fibrant object of G -Spectra G -global projective . G - I -simplicial sets. We will nowexhibit another connection between ( G -)spectra and ( G -) I -simplicial sets. For thiswe first recall the tensoring of G -Spectra over G - I -SSet ∗ , see e.g. [ Scha , Exam-ple 2.31] where this is denoted ‘– ∧ –’: .2. CONNECTIONS TO UNSTABLE G -GLOBAL HOMOTOPY THEORY 191 Construction . Let X be a spectrum and let Y be a I -simplicial set.We write X ⊗ Y for the spectrum with ( X ⊗ Y )( A ) = X ( A ) ∧ Y ( A ) and structuremaps S B r i ( A ) ∧ (cid:0) X ( A ) ∧ Y ( A ) (cid:1) ∼ = (cid:0) S B r i ( A ) ∧ X ( A ) (cid:1) ∧ Y ( A ) σ ∧ Y ( i ) −−−−→ X ( B ) ∧ Y ( B )for any injection i : A → B of finite sets, where the unlabelled isomorphism is theassociativity constraint. The tensor product becomes a simplicially enriched func-tor in both variables by applying the enriched functoriality of the smash productof pointed simplicial sets levelwise. If G is any discrete group, then taking thediagonal of the two actions promotes the tensor product to a simplicially enrichedfunctor G -Spectra × G - I -SSet ∗ → G -Spectra that we denote by the same sym-bol. Finally, precomposing with the forgetful functor we get a simplicially enrichedfunctor(3.2.2) G -Spectra × G - I -SSet ∗ → G -Spectra that we again denote by – ⊗ –. Remark . It is not hard to verify that these indeed become actions of G - I -SSet ∗ and G - I -SSet ∗ , respectively, on G -Spectra (with respect to the lev-elwise smash product) in a preferred way; we leave the details to the curious reader.3.2.2.1. Homotopical properties of the tensor product.
It will be natural to firststudy these equivariantly, so let us fix a finite group H . We begin with a comparisonof homotopy groups: Lemma . Let X ∈ H -Spectra and Y ∈ H - I -SSet ∗ . Then there existsa natural isomorphism π H ∗ ( X ⊗ Y ) ∼ = π H ∗ ( X ∧ Y ( U )) . Proof.
This is a standard argument, see e.g. [
Schb , Proposition 5.14]. Wewill only prove the claim for π H ; the general case is done analogously, but requiresmore notation. For this we consider π H ∗ ( X ⊗ Y ) = colim A ∈ s ( U ) [ S A , ( X ⊗ Y )( A )] H ∗ = colim A ∈ s ( U ) [ S A , X ( A ) ∧ Y ( A )] H ∗ ∼ = colim ( A,B ) ∈ s ( U ) [ S A , X ( A ) ∧ Y ( B )] H ∗ ∼ = colim A ∈ s ( U ) colim B ∈ s ( U ) [ S A , X ( A ) ∧ Y ( B )] H ∗ ∼ = colim A ∈ s ( U ) [ S A , X ( A ) ∧ colim B ∈ s ( U ) Y ( B )] H ∗ ∼ = colim A ∈ s ( U ) [ S A , X ( A ) ∧ Y ( U )] H ∗ = π H ∗ ( X ∧ Y ( U )) . Here the first isomorphism uses that the diagonal s ( U ) → s ( U ) is cofinal, thesecond one is the Fubini Theorem for colimits, and the third one uses that S A iscompact (in the derived sense) and that ∧ is cocontinuous in each variable. As allof the above isomorphisms are clearly natural, this finishes the proof. (cid:3) For the rest of this section we focus on (3 . .
2) as it is more tractable:
Proposition . Let X ∈ H -Spectra , Y ∈ H - I -SSet ∗ . Then the maps X ⊗ Y → X ⊗ Y ( U ∐ –) ← X ⊗ const Y ( U ) = X ∧ Y ( U ) induced by the inclusions A ֒ → U ∐ A ← ֓ U are π ∗ -isomorphisms.
92 3. STABLE G -GLOBAL HOMOTOPY THEORY Proof.
Replacing H by a subgroup if necessary, it suffices to show this for π H ∗ . By the previous lemma it suffices to prove that the induced maps π H ∗ ( X ∧ Y ( U )) → π H ∗ ( X ∧ Y ( U ∐ –)( U )) ← π H ∗ ( X ∧ (const Y ( U ))( U ))are isomorphisms. But under the canonical isomorphisms Y ( U ∐ –)( U ) ∼ = Y ( U ∐ U )(see Lemma 1.3.9) and (const Y ( U ))( U ) ∼ = Y ( U ), these are induced from the twoinclusions U ⇒ U ∐ U , so Lemma 1.3.11 implies that the maps Y ( U ) ⇒ Y ( U ∐ U )are H -equivariant weak equivalences. The claim follows immediately. (cid:3) Proposition . Let X be an H -spectrum and let Y be any H - I -simplicialset. Then the map ψ : X ∧ Σ • Y → X ⊗ Y associated to the bimorphism given in degree A, B by X ( A ) ∧ (Σ • Y )( B ) = X ( A ) ∧ S B ∧ Y ( B ) twist −−−→ S B ∧ X ( A ) ∧ Y ( B ) σ B,A ∧ Y ( B֒ → A ∐ B ) −−−−−−−−−−−−→ X ( B ∐ A ) ∧ Y ( A ∐ B ) X (twist) ∧ id −−−−−−−−→ X ( A ∐ B ) ∧ Y ( A ∐ B ) = ( X ⊗ Y )( A ∐ B ) is natural in both variables. Moreover, if X is flat, then ψ is a π ∗ -isomorphism. Proof.
The naturality is obvious. For the second statement, we observe that X ∧ Σ • Y X ⊗ YX ∧ Σ • Y ( U ∐ –) X ⊗ Y ( U ∐ –) X ∧ Σ • const Y ( U ) X ⊗ const Y ( U ) ψψψ commutes by naturality, where the vertical arrows are induced from the zig-zag inProposition 3.2.11 (using that Σ • = S ⊗ –). In particular, the proposition tells usthat the right hand vertical maps are π ∗ -isomorphisms and as X ∧ – preserves π ∗ -isomorphisms by [ Hau17 , Proposition 6.2-(i)], so are the left hand vertical maps.But the lower horizontal map in the above diagram literally agrees with thecanonical comparison map X ∧ Σ ∞ Y ( U ) → X ∧ Y ( U ), so it is even an isomorphism,see e.g. [ Scha , Proposition 3.5]. The claim follows by 2-out-of-3. (cid:3)
We can now use this to establish G -global properties of the tensor product: Proposition . Let X be a flat G -spectrum, and let Y be any G - I -simplicial set. Then the above map ψ : X ∧ Σ • Y → X ⊗ Y is a G -global equivalence. Moreover, if Y is cofibrant in the G -global model struc-ture, then α ! ( ψ ) is a G ′ -global weak equivalence for any homomorphism α : G → G ′ . Proof.
For the first statement we let ϕ : H → G be any homomorphism. Then ϕ ∗ ( ψ ) literally agrees with ψ : ( ϕ ∗ X ) ∧ Σ • ( ϕ ∗ Y ) → ( ϕ ∗ X ) ⊗ ( ϕ ∗ Y ). As flatness isa property of the underlying non-equivariant spectrum, ϕ ∗ X is again flat, so theclaim follows from the previous proposition. .2. CONNECTIONS TO UNSTABLE G -GLOBAL HOMOTOPY THEORY 193 For the second statement we observe that X ∧ Σ • Y is cofibrant in the G -globalprojective model structure by Proposition 3.2.2 together with Proposition 3.1.59, so G acts levelwise freely outside the base point on it by Remark 3.1.20. Moreover, G also acts levelwise freely outside the base point on Y , hence also on X ⊗ Y . The claimtherefore follows from the first statement together with Proposition 3.1.49. (cid:3) Proposition . The simplicially enriched functor – ⊗ – : G -Spectra × G - I -SSet ∗ → G -Spectra is a left Quillen bifunctor with respect to the G -global injective model structures. Proof.
It is obvious that – ⊗ – preserves colimits in each variable, so it sufficesto verify the pushout product axiom.For this let i be an injective cofibration (i.e. levelwise injection) of G -spectraand let j be an injective cofibration of G - I -simplicial sets. It is then obvious fromthe pushout product axiom in SSet ∗ that i (cid:3) j is again a cofibration, so it onlyremains to show that this is also a weak equivalence provided that i or j is. Asbefore it suffices for this that X ⊗ – is homotopical for every G -spectrum X andthat – ⊗ Y is homotopical for every G - I -simplicial set.We will prove the second statement—the proof of the first one is similar, buteasier. We let ϕ : H → G be any homomorphism from a finite group to G ; wethen have to show that ϕ ∗ (– ⊗ Y ) = ϕ ∗ (–) ⊗ ϕ ∗ Y sends G -global weak equiv-alences to H -equivariant weak equivalences, for which it is enough that – ⊗ Z preserves H -equivariant weak equivalences for any Z ∈ H - I -SSet ∗ . But indeed,Proposition 3.2.11 applied to a complete H -universe U provides a zig-zag of π ∗ -isomorphisms (and hence in particular of H -equivariant weak equivalences) between– ⊗ Z and – ∧ Z ( U ). By 2-out-of-3 it therefore suffices that the latter is homotopical.While this can in fact be checked by hand, we here simply observe that it is a specialcase of [ Hau17 , Proposition 6.2-(i)] as – ∧ Z ( U ) is isomorphic to – ∧ Σ ∞ Z ( U ) andsince all suspension spectra of G -simplicial sets are flat. (cid:3) G -global mapping spaces. We now introduce a two-variable adjoint ofthe above tensor product:
Construction . We define a simplicially enriched functor F I : G -Spectra op × G -Spectra → G - I -SSet ∗ as follows: if X and Y are G -spectra and A is a finite set, then F I ( X, Y )( A ) = maps( X ⊗ I ( A, –) + , Y )(we remind the reader that maps consists of all not necessarily G -equivariant mapsand that it is a G -simplicial set by conjugation, with based point coming from theconstant map). The covariant functoriality in A is given by contravariant functori-ality of I ( A, –); the contravariant functoriality in X and the covariant functorialityin Y are the obvious ones.We define a map ǫ X,Y : X ⊗ F I ( X, Y ) → Y as follows: in degree A , ǫ X,Y isgiven by the composition X ( A ) ∧ maps( X ⊗ I ( A, –) + , Y ) → X ( A ) ∧ maps( X ( A ) ∧ I ( A, A ) + , Y ( A )) → X ( A ) ∧ I ( A, A ) + ∧ maps( X ( A ) ∧ I ( A, A ) + , Y ( A )) → Y ( A )
94 3. STABLE G -GLOBAL HOMOTOPY THEORY where the first map is induced by evaluating in degree A , the second one sends x ∈ X n to x ∧ (id A , . . . , id A ), and the final one is the counit of the usual smash-homadjunction in SSet ∗ . We omit the easy verification that ǫ X,Y is a map of G -spectraand that it is natural (in the enriched sense) in both variables.We further define η X,Y : Y → F I ( X, X ⊗ Y ) as follows: in degree A , η X,Y is themap Y ( A ) → maps( X ⊗ I ( A, –) + , X ⊗ Y ) whose postcompisition with evaluationat B is the map Y ( A ) → maps( X ( B ) ∧ I ( A, B ) + , X ( B ) ∧ Y ( B )) adjunct to Y ( A ) ∧ X ( B ) ∧ I ( A, B ) + ∼ = Y ( A ) ∧ I ( A, B ) + ∧ X ( B ) → Y ( B ) ∧ X ( B )where the first map switches the factors and the second one comes from the structuremap of the simplicial functor Y . We again omit the straight-forward verificationthat this is well-defined and a simplicially enriched natural transformation. Wemoreover omit that for any fixed G -spectrum X the natural transformation ǫ X, – and η X, – satisfy the triangle equations, so that they form unit and counit of asimplicially enriched adjunction X ⊗ – : G - I -SSet ∗ ⇄ G -Spectra : F I ( X, –) . Clearly, F I (– , Y ) sends colimits to limits for any Y ∈ G -Spectra , so Proposi-tion 3.2.14 implies by adjointness: Corollary . The functor F I is a right Quillen bifunctor with respect tothe G -global injective model structures. (cid:3) Let
X, Y be G -spectra. Then there is also another way to cook up a ‘mapping G - I -space’ between X and Y , namely maps I ( I , Ω • F ( X, Y )). Our goal for the restof this section is to compare these two; more generally, we will need a version wherewe allow X to be an H -spectrum and we want a ( G × H )-global comparison. Construction . If T is any spectrum, then we have by adjointness andthe simplicially enriched Yoneda Lemma a sequence of isomorphismsmaps I ( I , Ω • T )( A ) ∼ = maps I ( I ( A, –) , maps I ( I , Ω • T )) ∼ = maps I ( I ( A, –) , Ω • T ) ∼ = maps(Σ • + I ( A, –) , T )natural in T and the finite set A . Applying this to T = F ( X, Y ) for X and Y spectra and appealing to the adjunction between ∧ and F we get an isomorphism(3.2.3) maps I (cid:0) I , Ω • F ( X, Y ) (cid:1) ( A ) ∼ = maps( X ∧ Σ • + I ( A, –) , Y )natural in X , Y , and A . In particular, if the group H acts on X and the group G acts on Y , then these maps for varying A assemble into an isomorphism in ( G × H )- I -SSet ∗ .We now define ˆ ψ : F I ( X, Y ) → maps I ( I , Ω • F ( X, Y )) as the map given in de-gree A by the compositionmaps( X ⊗ I ( A, –) + , Y ) ψ ∗ −−→ maps( X ∧ Σ • + I ( A, –) , Y ) ∼ = maps I (cid:0) I , Ω • F ( X, Y ) (cid:1) ( A )where the unlabelled isomorphism is (3 . . .3. G -GLOBAL SPECTRA VS. G -EQUIVARIANT SPECTRA 195 Theorem . If X is a flat H -spectrum and Y is fibrant in the G -globalinjective model structure, then (3.2.4) ˆ ψ : F I ( X, Y ) → maps I (cid:0) I , Ω • F ( X, Y ) (cid:1) is a ( G × H ) -global weak equivalence. Proof.
We show that it is a ( G × H )-global level equivalence. For this welet K be any finite group acting faithfully on A and ϕ : K → G × H any grouphomomorphism. We have to show that ϕ ∗ ( ˆ ψ )( A ) K is a weak homotopy equivalence,for which it is enough by construction of ˆ ψ that ϕ ∗ ( ψ ∗ )( A ) K is. But if we write ϕ : K → G and ϕ : K → H for the components of ϕ , then this agrees withmaps K ( ψ, ϕ ∗ Y ) : maps K ( ϕ ∗ X ⊗ I ( A, –) + , ϕ ∗ Y ) → maps K ( ϕ ∗ X ∧ Σ • + I ( A, –) , ϕ ∗ Y ) . Using that ϕ is a simplicial left adjoint to ϕ ∗ we see that this agrees up to conju-gation by isomorphisms with (cid:0) ϕ ( ψ ) (cid:1) ∗ : maps G ( ϕ ( ϕ ∗ X ⊗ I ( A, –) + ) , Y ) → maps G ( ϕ ( ϕ ∗ X ∧ Σ • + I ( A, –)) , Y ) . But ϕ ( ψ ) is a G -global weak equivalence by Proposition 3.2.13, and Y is fibrantin the injective G -global model structure by assumption. The claim follows sincethe latter is a simplicial model structure. (cid:3) G -global spectra vs. G -equivariant spectra In this section we will finally prove as promised:
Proposition . Let G be a finite group. Then the simplicial adjunctions (3.3.1) id : G -Spectra G -equivariant projective ⇄ G -Spectra G -global flat : id and (3.3.2) Σ • + I ( A, –) ∧ – : G -Spectra G -equiv. proj. ⇄ G -Spectra G -global proj. : Ω A sh A are Quillen adjunctions. In particular, if X is fibrant in the G -global flat modelstructure, then X is a G - Ω -spectrum.In addition, the adjunctions induced by (3 . . and (3 . . on associated quasi-categories are canonically equivalent and moreover right Bousfield localizations atthe G -equivariant weak equivalences. The key ingredient is the following consequence of the results of Subsection 3.1.5:
Proposition . Let A be a finite faithful G -set. Then (3.3.3) Σ • I ( A, –) ∧ – : G -Spectra G -global flat ⇄ G -Spectra G -global proj. : Ω A sh A is a simplicial Quillen equivalence. Moreover, the adjunction induced on associatedquasi-categories is canonically equivalent to the identity adjunction. Proof.
Forgetting about the action on A , we have for any spectra X , Y asequence of SSet -enriched natural isomorphismsmaps(Σ • + I ( A, –) ∧ X, Y ) ∼ = maps(Σ • + I ( A, –) , F ( X, Y )) ∼ = maps( I ( A, –) , Ω • F ( X, Y )) ∼ = Ω A maps( X, sh A Y ) ∼ = maps( X, Ω A sh A Y )where we have used in this order: the smash-function spectrum adjunction; theadjunction Σ • + ⊣ Ω • ; the SSet -enriched Yoneda Lemma together with the definitionof Ω • and F ; the fact that Ω A is defined in terms of the cotensoring of Spectra
96 3. STABLE G -GLOBAL HOMOTOPY THEORY over SSet ∗ . If G acts on X , Y , and A , then these isomorphisms are G -equivariantwith respect to the conjugate action by naturality, and taking G -fixed points of thisthus witnesses that (3 . .
3) is indeed a simplicial adjunction.As A is faithful, I ( A, –) ∼ = I ( A, –) × id G is cofibrant in the G -global modelstructure on G - I -SSet , and hence Σ • + I ( A, –) is G -globally projective by Proposi-tion 3.2.2; thus, (3 . .
3) is a Quillen adjunction by Proposition 3.1.59.To finish the proof it suffices to provide an equivalence R Ω A sh A ≃ id, for whichit is enough to give a natural levelwise weak equivalence betweenΩ A sh A : ( G -Spectra G -global projective ) f → G -Spectra G -global flat and the respective inclusion. A preferred such choice is provided by the maps λ from the proof of Lemma 3.1.34. (cid:3) Proof of Proposition 3.3.2.
Let us first show that the simplicial adjunc-tion (3 . .
1) is a Quillen adjunction. Composing with the simplicial Quillen adjunc-tion from Proposition 3.3.2 will then show that the same holds true for (3 . . X fibrant in the G -global flat model structure is a G -Ω-spectrum.By the proof the previous proposition, λ : X → Ω A sh A X is a weak equivalence,and both source and target are fibrant in the G -global flat model structure, soBrown’s Factorization Lemma asserts that λ factors as λ = ps where p is an acyclicfibration and s is a section of an acyclic fibration. As acyclic fibrations in the G -global flat model structure are in particular strong level equivalences, we concludethat also λ is a strong level equivalence. As G -Ω-spectra are obviously closed underthese, it then suffices that Ω A sh A X is a G -Ω-spectrum, which is obvious from theexplicit characterization of the fibrant objects provided by Theorem 3.1.36. Thiscompletes the proof that (3 . .
1) is a Quillen adjunction.Proposition 3.3.2 already implies that the induced adjunctions are canonicallyequivalent, so it only remains to show that (3 . .
1) induces a right Bousfield local-ization at the G -equivariant weak equivalences. This is however obvious becauseid ∞ = R id : G -Spectra ∞ G -global flat → G -Spectra ∞ G -equivariant projective is a quasi-localization at these by 2-out-of-3. (cid:3) On the other hand one immediately concludes from the definitions:
Corollary . Let G be a finite group. Then the simplicial adjunction id : G -Spectra G -global flat ⇄ G -Spectra G -equivariant flat : id . is a Quillen adjunction. The induced adjunction on associated quasi-categories is a(left) Bousfield localization at the G -equivariant weak equivalences. (cid:3) Definition . Let ϕ : H → G be any group homomorphism from a finitegroup H . We writeu ϕ : G -Spectra ∞ G -global → H -Spectra ∞ H -equivariant for the functor induced by the homotopical functor ϕ ∗ : G -Spectra G -global → H -Spectra H -equivariant .As an upshot of the above we conclude: .4. DELOOPING AND GROUP COMPLETION 197 Corollary . Let ϕ : H → G be as above. Then the simplicial adjunctions (3.3.4) ϕ ∗ : G -Spectra G -global flat ⇄ H -Spectra H -equivariant flat : ϕ ∗ and ϕ ! (Σ • + I ( A, –) ∧ –) : H -Spectra H -equiv. proj. ⇄ G -Spectra G -gl. proj. : Ω A sh A ϕ ∗ are Quillen adjunctions, where A is any finite faithful H -set (for example, A = H with H -action by left translation). In particular, u ϕ admits both a left and a rightadjoint. For G = 1, this (or rather its analogue in the world of orthogonal spectraand with respect to all compact Lie groups) is sketched in [ Sch18 , Remark 4.5.25]and an alternative proof of the existence of adjoints on the level of the homotopycategories is spelled out as Theorem 4.5.24 of op.cit.
Proof.
It is immediate from Lemma 3.1.45 together with Corollary 3.3.3 that ϕ ∗ in (3 . .
4) is left Quillen, proving the first statement and in particular providinga right adjoint of u ϕ .For the second statement it suffices to factor this as H -Spectra H -equiv. proj. H -Spectra H -gl. proj. G -Spectra G -gl. proj.Σ • + I ( A, –) ∧ –Ω A sh A ϕ ! ϕ ∗ and then appeal to Proposition 3.3.1 and Lemma 3.1.44. Finally, Proposition 3.3.2identifies the right adjoint of the induced adjunction with u ϕ , which shows that L ϕ ! (Σ • + I ( A, –) ∧ –) is the desired left adjoint. (cid:3) Segal’s classical Delooping Theorem [
Seg74 , Proposition 3.4] exhibits the ho-motopy theory of connective spectra as an explicit Bousfield localization of thehomotopy theory of Γ-spaces, also see [
BF78 , Theorem 5.8] for a model categori-cal formulation. A G -equivariant version of this (for any finite group G ) appearedas [ Ost16 , Theorem 6.5]. In this section we will strengthen this to a G -globalcomparison for any discrete group G . We begin by brieflyrecalling non-equivariant and equivariant deloopings of Γ-spaces.
Construction . The restriction
Fun ( SSet ∗ , SSet ∗ ) → Γ-SSet ∗ alongthe evident embedding Γ ֒ → SSet ∗ admits a fully faithful left adjoint via SSet ∗ -enriched Kan extension. We call this the prolongation of X . By full faithfulness,the prolongation of X agrees with X on Γ up to canonical isomorphism, so we willnot distinguish it notationally from the original Γ-space.Explicitly: let X ∈ Γ-SSet ∗ and let K be any pointed simplicial set. Then wedefine X ( K ) as the SSet ∗ -enriched coend (or, equivalently, SSet -enriched coend) Z S + ∈ Γ F ( S + ) × K S ;here the contravariant functoriality of S + K S is induced by the canonical iden-tification of the right hand side with the simplicial set maps( S + , K ) of base-pointpreserving maps.
98 3. STABLE G -GLOBAL HOMOTOPY THEORY Using the functoriality of coends, any map of K → L of simplicial sets induces X ( K ) → X ( L ), so this yields a SSet ∗ -enriched functor SSet ∗ → SSet ∗ ; similarly,any map X → Y of Γ-spaces induces a map X ( K ) → Y ( K ) for any K ∈ SSet ∗ .Now let G be a finite group. If X is a Γ- G -space, and K is any pointed G -simplicial set, then we make X ( K ) into a pointed G -simplicial set via the diagonalof the two actions.It is a classical observation that for finite K there exists a natural isomorphismbetween X ( K ) and the diagonal of the bisimplicial set ( m, n ) X ( K m ) n (with theevident functoriality in m and n ), see e.g. [ Sch99 , p. 331]. The equivariant DiagonalLemma (1.2.51) therefore immediately implies, also see [
Ost16 , Lemma 4.8]:
Corollary . (1) If X → Y is a G -equivariant level weak equiv-alence, then the induced map X ( K ) → Y ( K ) is a G -equivariant weakequivalence for any finite pointed G -simplicial set. (2) If K → L is any G -equivariant weak equivalence of finite pointed G -simplicial sets, then the induced map X ( K ) → X ( L ) is a G -equivariantweak equivalence for any Γ - G -space X . (cid:3) In fact, we can get rid of the finiteness assumption by filtering K (and L )appropriately, but we will only need the above version. Construction . We recall that a (symmetric) spectrum Y is a SSet ∗ -enriched functor Σ → SSet ∗ . As the prolongation of any Γ-space X is SSet ∗ -enriched, we can therefore define X ( Y ) := X ◦ Y : Σ → SSet ∗ . This becomes afunctor in X and Y in the obvious way.If we take the convention that coends and limits in functor categories are com-puted pointwise, the above literally agrees with the SSet ∗ -enriched coend Z S + ∈ Γ X ( S + ) ∧ Y S Y. We can also describe X ( Y ) explicitly as follows: if B is any finite set, then X ( Y )( B ) = X ( Y ( B )) with the evident Σ B -action; for any further finite set A , the structure map S A ∧ X ( Y )( B ) = S A ∧ X ( Y ( B )) → X ( Y ( A ∐ B )) = X ( Y )( A ∐ B ) is given by thecomposite S A ∧ X ( Y ( B )) asm −−→ X ( S A ∧ Y ( B )) X ( σ ) −−−→ X ( Y ( B ))where the left hand arrow is the assembly map induced by the SSet ∗ -enrichmentof the prolongation.Again, we can evaluate any Γ- G -space X at any G -spectrum by applying theabove construction and then pulling through the actions. Definition . The associated spectrum of a Γ- G -space X is the G -spectrum X ( S ). We write E G : Γ- G -SSet ∗ → G -Spectra for the functor X X ( S ).Note that X ( S )( ∅ ) = X ( S ) ∼ = X (1 + ) is the ‘underlying G -space’ of the Γ- G -space X . For suitable X , we can think of the remaining data as an equivariantdelooping of this G -space: Definition . A Γ- G -space is called very special if it is special and themonoid π H ( X (1 + )) from Remark 2.2.12 is a group for all H ⊂ G . Theorem . If X is very special, then X ( S ) is a G - Ω -spectrum. .4. DELOOPING AND GROUP COMPLETION 199 Proof.
See e.g. [
Ost16 , Proposition 5.7]. (cid:3) [ Ost16 , Theorem 6.2] introduces a stable model structure on Γ- G -SSet ∗ withthe same cofibrations as the G -equivariant level model structure (see Proposi-tion 2.2.33), and whose fibrant objects are precisely the very special level fibrantΓ- G -spaces. Proposition . The functor E G is part of a simplicial Quillen adjunction E G : ( Γ- G -SSet ∗ ) stable ⇄ G -Spectra G -equivariant : Φ G . Proof. [ Ost16 , Proposition 5.2] shows this for the level model structures; theclaim follows as the right adjoint sends fibrant objects to very special Γ- G -spacesby [ Ost16 , proof of Theorem 5.9]. (cid:3)
Together with Corollary 3.4.2 we conclude that E G is in fact homotopical. Proposition . The G -spectrum X ( S ) is connective , i.e. π ∗ X ′ vanishes innegative degrees for some (hence any) G -equivariant weak equivalence X ( S ) → X ′ to a G - Ω -spectrum X ′ . Proof. [ Ost16 , Corollary 5.5] shows that the negative na¨ıve homotopy groupsof Y ( S ) vanish for any Y ∈ Γ- G -SSet ∗ .To prove the proposition, we now simply choose the replacement X ( S ) → X ′ asthe image under E G of a fibrant replacement X → Y in the stable model structureon Γ- G -SSet ∗ ; here we used that X ′ → Y is indeed a weak equivalence as E G ishomotopical, and that X ′ = Y ( S ) is a G -Ω-spectrum by Theorem 3.4.6. (cid:3) Remark . One can in fact show that the na¨ıve homotopy groups π ∗ ( X ( S ))agree with the homotopy groups of X ′ (i.e. X ( S ) is semistable ), but we will notneed this below.The key result on the homotopy theory of very special Γ-spaces is the following: Theorem . The adjunction ( E G ) ∞ ⊣ R Φ G re-stricts to a Bousfield localization ( E G ) ∞ : ( Γ- G -SSet ∗ ) ∞ level w.e. ⇄ G -Spectra ∞≥ : R Φ G where G -Spectra ≥ denotes the subcategory of connective G -spectra. Moreover,the essential image of R Φ G consists precisely of the very special Γ - G -spaces. Proof.
This is immediate from the model categorical statement proven in[
Ost16 , Proposition 5.2 and Theorem 6.5], also see [
BF78 , Theorem 5.8] for thespecial case G = 1. (cid:3) G -Global Delooping Theorem. Before we can properly statea G -global version of the Delooping Theorem, we first need to understand what itmeans to be ‘connective’ or ‘very special’ in the G -global context. Definition . A G -spectrum X is G -globally connective if ϕ ∗ X is H -equivariantly connective for all finite groups H and all homomorphisms ϕ : H → G . Definition . We call X ∈ Γ- G - I -SSet ∗ very special if u ϕ X is veryspecial for every finite group H and any homomorphism ϕ : H → G .
00 3. STABLE G -GLOBAL HOMOTOPY THEORY Put differently (see Lemma 2.2.48), X is very special if and only if it specialand the abelian monoid structure on π H (( ϕ ∗ X )(1 + )( U )) induced by X (1 + ) × X (1 + ) X (2 + ) X (1 + ) ∼ ρ X ( µ ) is a group structure for every finite group H and any homomorphism ϕ : H → G .Next we introduce the G -global delooping functor, cf. [ Sch19b , Construc-tion 3.3]:
Construction . We define E ⊗ : Γ- G - I -SSet ∗ → G -Spectra via the SSet ∗ -enriched coend E ⊗ ( X ) = Z T + ∈ Γ Y T S ! ⊗ X ( T + ) . together with the evident SSet ∗ -enriched functoriality. Remark . We again take the convention that the above coend is con-structed by forming the levelwise coend. In this case, E ⊗ ( X )( A ) = Z T + ∈ Γ Y T S A ! ∧ X ( T + )( A ) = X ( S A )( A ) , and the structure map E ⊗ ( X )( B ) → E ⊗ ( X )( A ∐ B ) is given by the diagonal com-posite S A ∧ X ( S B )( B ) X ( S A ∧ S B )( B ) X ( S A ∐ B )( B ) S A ∧ X ( S B )( A ∐ B ) X ( S A ∧ S B )( A ∐ B ) X ( S A ∐ B )( A ∐ B ) . S A ∧ X ( S B )( B֒ → A ∐ B ) asm ∼ = X ( S A ∐ B )( B֒ → A ∐ B )asm ∼ = Comparison to equivariant deloopings.
We now want to relate this toShimakawa’s equivariant version of Segal’s machinery.
Remark . If X is any Γ- G -space, then E ⊗ (const X ) = E G ( X ). Moreprecisely, the diagram Γ- G -SSet ∗ G -SpectraΓ- G - I -SSet ∗ const E G E ⊗ of simplicially enriched functors is strictly commutative. Corollary . Let X ∈ Γ- G - I -SSet ∗ be fibrant in the injective G -globalmodel structure. Then the degree zero inclusion induces a G -equivariant level weakequivalence E G ( X ( ∅ )) = E ⊗ (const X ( ∅ )) → E ⊗ ( X ) . Proof.
We have to show that the inclusion induces a G -equivariant weakequivalence X ( ∅ )( S A ) → X ( A )( S A ) for every finite G -set A . But X ( ∅ ) → X ( A )is a G -equivariant level equivalence of Γ- G -spaces by Corollary 2.2.40, so the claimfollows from Corollary 3.4.2. (cid:3) Corollary . Let G be finite and assume X ∈ Γ- G - I -SSet ∗ is fibrantin the injective G -global model structure and moreover G -globally very special. Then E ⊗ ( X ) is a G - Ω -spectrum. .4. DELOOPING AND GROUP COMPLETION 201 Proof.
By the previous corollary, E ⊗ ( X ) is G -equivariantly level equivalentto E G ( X ( ∅ )). The claim now follows from Theorem 3.4.6 as X ( ∅ ) is very specialby Corollary 2.2.41. (cid:3) For a general comparison we introduce:
Construction . Let H be a finite group and let ϕ : H → G be a ho-momorphism. We define E ϕ : Γ- G - I -SSet ∗ → H -Spectra as follows: if A is anyfinite set and X is any Γ- G - I -space, then E ϕ ( X )( A ) = ( ϕ ∗ X )( U ∐ A )( S A ), where U is the complete H -set universe from the definition of the underlying H -simplicialset of a G - I -simplicial set. The structure maps of E ϕ ( X ) are defined analogouslyto Remark 3.4.14 and the functoriality in X is the obvious one.We now consider the zig-zag of natural transformations(3.4.1) ϕ ∗ ◦ E ⊗ ⇒ E ϕ ⇐ E H ◦ ev U ◦ ϕ ∗ induced in degree A by the inclusions A ֒ → U ∐ A ← ֓ U .The following in particular generalizes (the simplicial analogue of) [ Sch19b ,Theorem 3.13]:
Lemma . Both maps in (3 . . are π ∗ -isomorphisms (and hence in par-ticular H -equivariant weak equivalences). Proof.
Analogously to Lemma 3.2.10 one gets natural isomorphisms π ∗ u ϕ E ⊗ ( X ) ∼ = π ∗ E H (cid:0) ( ϕ ∗ X )( U ) (cid:1) and π ∗ ( E ϕ X ) ∼ = π ∗ E H (cid:0) ( ϕ ∗ X )( U ∐ U )) , for every X ∈ Γ- G - I -SSet ∗ , and under this identification the actions of the maps(3 . .
1) on homotopy groups are induced by the two inclusions U ֒ → U ∐ U ← ֓ U .Thus, it is enough to show that the induced maps ( ϕ ∗ X )( U ) → ( ϕ ∗ X )( U ∐ U )are H -equivariant weak equivalences of Γ- H -spaces, i.e. if T is any finite H -set,then the induced maps ( ϕ ∗ X )( U )( T + ) → ( ϕ ∗ X )( U ∐ U )( T + ) are H -equivariantweak equivalences. But this is simply an instance of Lemma 1.3.11 applied to the H - I -simplicial set ( ϕ ∗ X )(–)( T + ). (cid:3) Corollary . The functor E ⊗ sends G -global level weak equivalences to G -global weak equivalences. Moreover, it takes values in G -globally connective G -spectra. (cid:3) Corollary . The diagram Γ- G - I -SSet ∞∗ Γ- H -SSet ∞∗ G -Spectra ∞ G -global H -Spectra ∞ H -equivariant( E ⊗ ) ∞ u ϕ E ∞ H u ϕ commutes up to preferred equivalence. (cid:3) Now we can finally state the main result of this section:
Theorem G -global Delooping Theorem) . Let G be any group. Then ( E ⊗ ) ∞ : ( Γ- G - I -SSet ∗ ) ∞ G -global → ( G -Spectra ≥ ) ∞ G -global (where the subscript ‘ ≥ ’ denotes the subcategory of G -globally connective spectra)admits a fully faithful right adjoint R Φ ⊗ , yielding a Bousfield localization. More-over, the following hold:
02 3. STABLE G -GLOBAL HOMOTOPY THEORY (1) The essential image of R Φ ⊗ consists precisely of the very special G -global Γ -spaces. (2) ( E ⊗ ) ∞ inverts a map f if and only if u ϕ f is a stable weak equivalence forall finite groups H and all homomorphisms ϕ : H → G . (3) For any finite group H and any homomorphism ϕ : H → G there existpreferred equivalences filling Γ- G - I -SSet ∞∗ Γ- H -SSet ∞∗ G -Spectra ∞ G -global H -Spectra ∞ H -equivariant( E ⊗ ) ∞ u ϕ E ∞ H u ϕ and Γ- G - I -SSet ∞∗ Γ- H -SSet ∞∗ G -Spectra ∞ G -global H -Spectra ∞ H -equivariantu ϕ u ϕ R Φ ⊗ R Φ H and these are moreover canonical mates of each other. G -global Γ -spaces from G -global spectra. The proof requires some prepa-rations and will occupy the rest of this subsection. We begin by introducing theright adjoint of ( E ⊗ ) ∞ , which already exists on the pointset level: Construction . We define Φ ⊗ : G -Spectra → Γ- G - I -SSet ∗ as fol-lows: if X is a G -spectrum and T is a finite set, then (Φ ⊗ X )( T + ) = F I ( Q T S , X ).The functoriality in T + is induced by the natural identification Q T S ∼ = maps( T + , S )and the SSet ∗ -enriched functoriality in X is the obvious one.We define for every G -spectrum X the map ǫ X : E ⊗ Φ ⊗ X → X as the oneinduced by the maps ǫ Q T S ,X : Y T S ! ⊗ F I Y T S , X ! → X for varying T + ∈ Γ, where ǫ comes from Construction 3.2.15. We omit the easyverification that this is well-defined and a SSet ∗ -enriched natural transformation.If Y is any G - I -space, we define η Y : Y → Φ ⊗ E ⊗ Y as the map given in degree T + by Y ( T + ) η −→ F I Y T S , Y T S ! ⊗ Y ( T + ) ! → F I Y T S , Z U + ∈ Γ Y U S ! ⊗ Y ( U + ) ! where the right hand arrow is induced by the structure map of the coend and theleft hand arrow is again from Construction 3.2.15. We omit the easy verificationthat also η is a well-defined simplicial transformation, and that ǫ and η satisfy thetriangle identities, yielding an enriched adjunction(3.4.2) E ⊗ : Γ- G - I -SSet ∗ ⇄ G -Spectra : Φ ⊗ . Remark . One immediately checks from the above definitions that thecounit ǫ X : E ⊗ Φ ⊗ X → X is an isomorphism in degree ∅ for any G -spectrum X . Corollary . The simplicial adjunction (3 . . is a Quillen adjunctionwith respect to the G -global injective model structures. .4. DELOOPING AND GROUP COMPLETION 203 Proof.
Using the description from Remark 3.4.14 it is obvious that E ⊗ pre-serves injective cofibrations. Moreover, it is homotopical by Corollary 3.4.20. (cid:3) We will now give an alternative description of the right adjoint that will bemore tractable.
Construction . Let us define Φ Σ : G -Spectra → Γ- G -Spectra viaΦ Σ ( X )( T + ) = F ( Q T S , X ). Here the functoriality in T + comes from the naturalidentification maps( T + , S ) ∼ = Q T S and the SSet ∗ -enriched functoriality in X is theobvious one.Using this, we now define Φ ∧ as the composite G -Spectra Φ Σ −−→ Γ- G -Spectra Ω • −−→ Γ- G - I -SSet ∗ maps I ( I , –) −−−−−−−→ Γ- G - I -SSet ∗ . Corollary . The maps ˆ ψ from Construction 3.2.17 define a naturaltransformation ˆ ψ : Φ ⊗ ⇒ Φ ∧ . If X is fibrant in the G -global injective model struc-ture, then ˆ ψ X : Φ ⊗ X → Φ ∧ X is a G -global level weak equivalence. Proof. As Q T S is a flat Σ T -spectrum for any finite set T (Example 3.1.9),this follows by applying Theorem 3.2.18 levelwise. (cid:3) Remark . Also Φ ∧ admits a simplicial left adjoint, which can be com-puted via the coend E ∧ ( X ) = R T + ∈ Γ ( Q T S ) ∧ Σ • X , and the resulting adjunction E ∧ ⊣ Φ ∧ is left Quillen with respect to the G -global level model structure on thesource and the G -global projective model structure on the target (this uses Propo-sition 3.1.59 again, also cf. the argument in the proof of the proposition below).However, we will not need this below, so we leave the details to the interestedreader. Proposition . Let X ∈ G -Spectra be fibrant in the G -global projectivemodel structure. Then Φ ∧ ( X ) is very special. Proof.
Let us first prove that Φ ∧ ( X ) is special, for which we let T be anyfinite set. Up to isomorphism, the Segal map Φ ∧ ( X )( T + ) → Q T Φ ∧ ( X )(1 + ) isgiven by(3.4.3) maps I I , Ω • F Y T S , X !! maps I ( I , Ω • c ∗ ) −−−−−−−−−→ maps I I , Ω • F _ T S , X !! , where c is the canonical map W T S → Q T S . The latter is a Σ T -global weak equiv-alence (Proposition 3.1.53) between flat Σ T -spectra (Examples 3.1.8 and 3.1.9, re-spectively); as X is fibrant in the G -global projective model structure, c ∗ is thereforea ( G × Σ T )-global weak equivalence between ( G × Σ T )-globally projectively fibrant( G × Σ T )-spectra by Corollary 3.1.61 and Ken Brown’s Lemma. As both Ω • andmaps I ( I , –) are right Quillen with respect to the corresponding projective modelstructures, we conclude that also (3 . .
3) is a ( G × Σ T )-global weak equivalence, andhence so is the Segal map.Now let ϕ : H → G be any homomorphism from a finite group to G and let U be the fixed complete H -set universe. By the above, ( ϕ ∗ Φ ∧ X )( U ) is a specialΓ- H -space and to finish the proof we have to show that the induced monoid struc-ture on π H ( ϕ ∗ Φ ∧ X ( U )(1 + )) = π H (cid:0) maps I ( I , Ω • ϕ ∗ F ( S , X ))( U ) (cid:1) is actually a group
04 3. STABLE G -GLOBAL HOMOTOPY THEORY structure. As in the classical setting, this is an application of the Eckmann-Hiltonargument: first, we consider the diagram(3.4.4) π H (cid:0) maps I ( I , Ω • ϕ ∗ F ( S , X ))( U ) (cid:1) π H ( ϕ ∗ F ( S , X )) π H (cid:0) maps I ( I , Ω • ϕ ∗ F ( S × S , X ))( U ) (cid:1) π H ( ϕ ∗ F ( S × S , X )) π H (cid:0) maps I ( I , Ω • ϕ ∗ F ( S , X ))( U ) (cid:1) × π H ( ϕ ∗ F ( S , X )) × p ∗ ,p ∗ ) ∼ = µ ∗ µ ∗ ( p ∗ ,p ∗ ) ∼ = (which commutes by naturality) where the horizontal arrows are those from Re-mark 3.2.7 applied to the H -spectra ϕ ∗ F ( S , X ) and ϕ ∗ F ( S × S , X ). As both ofthese are fibrant in the H -global projective model structure by Proposition 3.1.59together with Lemma 3.1.44, the remark tells us that the horizontal arrows areactually bijective.By commutativity, we then conclude that the right hand vertical arrows equip π H ( ϕ ∗ F ( S , X )) with the structure of an abelian monoid, and that the top hori-zontal arrow in (3 . .
4) is an isomorphism with respect to this monoid structure.But on the other hand, the vertical arrows on the right are group homomor-phisms with respect to the usual group structure on π H , so the Eckmann-Hiltonargument implies that this monoid structure on π H ( ϕ ∗ F ( S , X )) agrees with thestandard group structure on π H . We conclude that also the monoid structureon π H (cid:0) maps I ( I , Ω • ϕ ∗ F ( S , X ))( U ) (cid:1) = π H ( ϕ ∗ Φ ∧ ( X )( U )(1 + )) is a group structure,which completes the proof. (cid:3) Remark . In appealing to Remark 3.2.7 the above proof secretly invokesthe equivalence between H - I -SSet and H - I -SSet (Theorem 1.3.31). Corollary . Let X ∈ G -Spectra be fibrant in the G -global model in-jective structure. Then Φ ⊗ ( X ) is very special. (cid:3) Proof of the Delooping Theorem.
The following will be the main stepin the proof of Theorem 3.4.22:
Proposition . Let X ∈ G -Spectra ≥ be fibrant in the G -global injectivemodel structure. Then the counit E ⊗ Φ ⊗ X → X is a G -global weak equivalence. Proof.
Let H be a finite group, let ϕ : H → G be a group homomorphism,and let i : ϕ ∗ X → X ′ be a fibrant replacement in the injective H -global modelstructure. Claim. Φ ⊗ ( i ) : Φ ⊗ ( ϕ ∗ X ) → Φ ⊗ ( X ′ ) is an H -global level weak equivalence. Proof.
By Corollary 3.4.31, the H -global Γ-space Φ ⊗ ( X ′ ) is special, and so isthe G -global Γ-space Φ ⊗ ( X ). It follows that also Φ ⊗ ( ϕ ∗ X ) = ϕ ∗ Φ ⊗ ( X ) is special.It is therefore enough to show that Φ ⊗ ( i )(1 + ) is an H -global weak equiva-lence (cf. Lemma 2.2.52). But this agrees up to conjugation by isomorphisms withmaps I ( I , Ω • ( i )). As both ϕ ∗ X and X ′ are in particular fibrant in the H -global projective model structure, the claim follows from Ken Brown’s Lemma. △ .4. DELOOPING AND GROUP COMPLETION 205 We have to show that ϕ ∗ ( ǫ X ) = ǫ ϕ ∗ X is an H -equivariant equivalence. But inthe naturality square E ⊗ Φ ⊗ ( ϕ ∗ X ) ϕ ∗ X E ⊗ Φ ⊗ ( X ′ ) X ′ ǫ ϕ ∗ X E ⊗ Φ ⊗ ( i ) iǫ X ′ the right hand vertical arrow is even an H -global weak equivalence, and so is theleft hand vertical arrow by the above claim together with Corollary 3.4.20. Thus,it is enough to show that ǫ X ′ is an H -equivariant weak equivalence.For this we observe that Φ ⊗ ( X ′ ) is fibrant in the injective H -global modelstructure on Γ- H - I -SSet ∗ by Corollary 3.4.25 and moreover very special by Corol-lary 3.4.31. Thus, Corollary 3.4.17 implies that E ⊗ Φ ⊗ ( X ′ ) is an H -Ω-spectrum. Inaddition, Corollary 3.4.20 implies that it is also H -globally connective.On the other hand, X ′ is an H -Ω-spectrum by Proposition 3.3.1 and it isconnective by assumption. As ǫ X ′ ( ∅ ) is an isomorphism, it follows immediatelythat ǫ X ′ is even a H -equivariant level equivalence, finishing the proof. (cid:3) Instead of the above approach, we could have compared the counit of ourdelooping adjunction to the counit of the usual equivariant one. However, we havecarefully avoided such a comparison because the computations involved can becomequite nasty. Instead, we will get all these compatibilities for free now thanks to thefollowing technical lemma:
Lemma . Let (3.4.5) F : C ⇄ D : U be an adjunction of quasi-categories, let I be a set (or more generally any class),and let ( S i : C → C i ) i ∈ I and ( T i : D → D i ) i ∈ I be two jointly conservative families.Assume moreover we are given for each i ∈ I a Bousfield localization F i : C i ⇄ D i : U i such that ess im S i U ⊂ ess im U i , together with a natural equivalence ϕ i filling (3.4.6) C C i D D i . S i F F iϕi ≃ ⇒ T i Then the following are equivalent: (1) U is fully faithful, i.e. also (3 . . is a Bousfield localization. (2) For all i ∈ I , the canonical mate S i U ⇒ U i T i of (3 . . is an equivalence.Moreover, in this case X ∈ D lies in the essential image of U if and only if S i X lies in the essential image of U i for all i ∈ I , and a map f is inverted by F if andonly if S i f is inverted by F i for all i ∈ I . Proof.
All the properties in question can be checked on the level of homotopycategories, so it suffices to prove this for ordinary 1-categories.
06 3. STABLE G -GLOBAL HOMOTOPY THEORY (1) ⇒ (2): If i ∈ I is arbitrary, then the canonical mate ¯ ϕ i of ϕ i is by definitiongiven by the pasting(3.4.7) D C C i D D i C iU = ǫ ⇒ S i F F iϕi ≃ ⇒ = T i η ⇒ U i Since U is fully faithful, ǫ is an isomorphism, and so is ϕ i by assumption. Onthe other hand, S i U X ∈ ess im U i for any X ∈ D by assumption, so that also η i evaluated at S i U X is an isomorphism because U i is fully faithful. We conclude thatthe above pasting is indeed an isomorphism as desired.(2) ⇒ (1): We have to show that the counit ǫ X of F ⊣ U is an equivalence forevery X ∈ D . As the T i ’s are jointly conservative, this is equivalent to T i ǫ X beingan isomorphism for all i ∈ I , and since U i is fully faithful, this is in turn equivalentto U i T i ǫ X being an isomorphism. But (2) precisely tells us that the composition S i U X ( η i ) SiUX −−−−−−→ U i F i S i U U I ( ϕ i ) UX −−−−−−→ U i T i F U X U i T i ǫ X −−−−−→ U i T i X is an isomorphism, and so are the first two arrows in it by the assumption oness im S i U and ϕ i , respectively. The claim follows by 2-out-of-3.It only remains to prove the characterizations of the maps inverted by F andthe essential image of U under these equivalent assumptions. But indeed, if f isany morphism in C , then F f is an isomorphism if and only if each T i F f is, whichby the isomorphism (3 . .
6) is equivalent to F i S i f being an isomorphism.Finally, if X lies in the essential image of U , then S i X ∈ ess im U i for all i ∈ I by assumption. For the converse, we observe that η X : X → U F X is inverted by F for all X ∈ C , i.e F i S i η X is an isomorphism for all i ∈ I by the above. But S i U F X ∼ = U i T i F X ∈ ess im U i ; as also S i X ∈ ess im U i by assumption, bothsource and target of S i η X lie in ess im U i , so S i η X is inverted by F i if and onlyif it is an isomorphism, i.e. S i η X is an isomorphism. Letting i vary and usingjoint conservativity again, we therefore conclude that η X is an isomorphism, hence X ∈ ess im U . This completes the proof of the lemma. (cid:3) We can now use the lemma to finally prove the G -global Delooping Theorem.It turns out to be convenient to prove this in parallel with the following result: Theorem . The inclusion ι : ( G -Spectra ≥ ) ∞ G -gl. ֒ → G -Spectra ∞ G -gl. ofthe G -globally connective G -spectra admits a right adjoint τ . For any homomor-phism ϕ the canonical mate u ϕ τ ⇒ τ u ϕ of the transformation ( G -Spectra ≥ ) ∞ G -global G -Spectra ∞ G -global ( H -Spectra ≥ ) ∞ H -equivariant H -Spectra ∞ H -equivariantu ϕ u ϕ ⇒ induced by the identity is an equivalence. Proof of Theorems 3.4.22 and 3.4.34.
We will break up the argumentinto several steps.
Step 1.
We prove a version of Theorem 3.4.22 for connective G -global/ H -equivariant spectra. For this we let I be the class of all homomorphisms ϕ : H → G .4. DELOOPING AND GROUP COMPLETION 207 from finite groups to G . We want to apply the lemma to the adjunction (3 . . E ∞ H : ( Γ- H -SSet ∗ ) ∞ H -equivariant ⇄ ( H -Spectra ≥ ) ∞ H -equivariant : R Φ H from Theorem 3.4.10 for all ϕ : H → G , and the jointly conservative families (cid:0) u ϕ : ( Γ- G - I -SSet ∗ ) ∞ G -global → ( Γ- H -SSet ∗ ) ∞ H -equivariant (cid:1) ϕ ∈ I and (cid:0) u ϕ : ( G -Spectra ≥ ) ∞ G -global → ( H -Spectra ≥ ) ∞ H -equivariant (cid:1) ϕ ∈ I . The equivalences (3 . .
6) are provided by Corollary 3.4.21, and Corollary 3.4.31asserts that u ϕ X is H -equivariantly very special for any very special G -global Γ-space and any ϕ : H → G . Finally, Proposition 3.4.32 verifies Condition (1) ofthe lemma. We may therefore conclude that the essential image of R Φ ⊗ consistsprecisely of the very special G -global Γ-spaces, that a map is inverted by ( E ⊗ ) ∞ if and only if it sent under u ϕ to a stable weak equivalence for all ϕ , and that thecanonical mates(3.4.9) ( Γ- G - I -SSet ∗ ) ∞ G -global ( G -Spectra ≥ ) ∞ G -global ( Γ- H -SSet ∗ ) ∞ H -equivariant ( H -Spectra ≥ ) ∞ H -equivariantu ϕ ⇒ R Φ ⊗ u ϕ R Φ H of the given equivalences are again weak equivalences. Note that this almost com-pletes the proof of Theorem 3.4.22 except that we want the equivalences (3 . . Step 2.
We will now prove Theorem 3.4.34. In order to avoid confusion, let usmomentarily write ( e ⊗ ) ∞ for ( E ⊗ ) ∞ viewed as a functor to ( G -Spectra ≥ ) ∞ G -global ,i.e. ( E ⊗ ) ∞ = ι ◦ ( e ⊗ ) ∞ . It is now a purely formal calculation (using the results of theprevious step) that τ := ( e ⊗ ) ∞ ◦ R Φ ⊗ is the desired right adjoint; the counit can betaken to be the counit of ( E ⊗ ) ∞ ⊣ R Φ ⊗ (which is indeed a natural transformationfrom ( E ⊗ ) ∞ ◦ R Φ ⊗ = ι ◦ ( e ⊗ ) ∞ ◦ R Φ ⊗ = ι ◦ τ to the identity). The rest of theclaim then follows by an easy application of the opposite of Lemma 3.4.33. Step 3.
We can now finish the proof of Theorem 3.4.22: we have to show thatthe canonical mate of the equivalence(3.4.10) ( Γ- G - I -SSet ∗ ) ∞ G -global G -Spectra ∞ G -global ( Γ- H -SSet ∗ ) ∞ H -equivariant H -Spectra ∞ H -equivariant( E ⊗ ) ∞ u ϕ u ϕ E ∞ H ⇒ from Corollary 3.4.21 is itself an equivalence. However, (3 . .
10) agrees (up tocanonical higher homotopy) with the pasting( Γ- G - I -SSet ∗ ) ∞ G -global ( G -Spectra ≥ ) ∞ G -global G -Spectra ∞ G -global ( Γ- H -SSet ∗ ) ∞ H -equiv. H -Spectra ∞ H -equiv. ( H -Spectra ≥ ) ∞ H -equiv.( E ⊗ ) ∞ u ϕ u ϕ u ϕ E ∞ H ⇒ ⇒ where the left hand square is filled with the restriction of the previous equivalence,and the right hand square is filled with the identity. By the compatibility of mates
08 3. STABLE G -GLOBAL HOMOTOPY THEORY with pasting, the canonical mate of (3 . .
10) is the pasting of the mates of these twosquares, and both of these were seen to be equivalences above. (cid:3)
Corollary . The homotopical functor E ⊗ induces a (Bousfield) local-ization (3.4.11) ( Γ- G - I -SSet special ∗ ) ∞ → G -Spectra ∞≥ . Proof.
By the above theorem, the fully faithful right adjoint R Φ ⊗ of E ∞ : ( Γ- G - I -SSet ∗ ) ∞ → G -Spectra ∞≥ factors through the very special G -global Γ-spaces, yielding a fully faithful rightadjoint of (3 . . (cid:3) HAPTER 4 G -global algebraic K -theory In this final chapter we construct G -global algebraic K -theory , generalizingSchwede’s global algebraic K -theory [ Sch19b ] and refining classical G -equivariantalgebraic K -theory. We then employ the theory of G -global infinite loop spacesdeveloped in the previous chapters to prove G -global refinements of Thomason’stheorem [ Tho95 , Theorem 5.1] that symmetric monoidal categories model all con-nective stable homotopy types.
We will give two equivalent constructions of G -global algebraic K -theory: oneas a straight-forward generalization of Schwede’s global algebraic K -theory, and theother one by adapting the usual construction of G -equivariant algebraic K -theory. G -global K -theory of G -parsummable categories. Let us beginby recalling the basic setup [
Sch19b , Definitions 2.11, 2.12, and 2.31] of Schwede’sapproach to global algebraic K -theory: Definition . A small E M -category C (i.e. a category with a strict actionof the categorical monoid E M ) is tame if its underlying M -set Ob( C ) of objectsis tame. The support supp( c ) of an object c ∈ C is its support as an element of the M -set Ob( C ).The box product C ⊠ D of two tame E M -categories is the full subcategory of C × D spanned by all pairs ( c, d ) such that supp( c ) ∩ supp( d ) = ∅ .We write E M -Cat τ for the category of tame E M -categories and strictly E M -equivariant functors. The box product then becomes a subfunctor of the cartesianproduct on E M -Cat τ , and [ Sch19b , Proposition 2.31] shows that the structureisomorphisms of the cartesian symmetric monoidal structure restrict to make ⊠ intothe tensor product of a preferred symmetric monoidal structure on E M -Cat τ .We can now define [ Sch19b , Definition 4.1]:
Definition . We write
ParSumCat := CMon( E M -Cat τ , ⊠ ) and callits objects the parsummable categories .The construction of the global algebraic K -theory of a parsummable categorycan be broken up into two steps: first, we construct a ‘global Γ-category,’ which wethen deloop to a global spectrum. Construction . Let us write Γ- E M -Cat ∗ for the category of functors X : Γ → E M -Cat with X (0 + ) = ∗ .Analogously to Construction 2.3.2, we can now define ̥ : ParSumCat → Γ- E M -Cat ∗ with ̥ ( X )( n + ) = X ⊠ n and the evident functoriality in X ; the G -GLOBAL ALGEBRAIC K -THEORY structure maps of X are again given by the universal property of ⊠ as coprod-uct in ParSumCat , also see [
Sch19b , Construction 4.3].
Construction . We obtain a functor Γ- E M -Cat τ ∗ → Γ- E M -SSet ∗ by applying the nerve levelwise.Now let X ∈ Γ- E M -SSet ∗ . The associated (symmetric) spectrum X h S i [ Sch19b , Construction 3.3] is given by X h S i ( A ) = X [ ω A ]( S A )(see Construction 1.3.21). If i : A → B is an injection, then the structure map S B r i ( A ) ∧ X h S i ( A ) → X h S i ( A ) is the composition S B r i ( A ) ∧ X [ ω A ]( S A ) asm −−→ X [ ω A ]( S A ∧ S B r i ( A ) ) ∼ = X [ ω A ]( S B ) X [ i ! ]( S B ) −−−−−−→ X [ ω B ]( S B ) . This becomes a functor in X in the obvious way.Put differently (see Remark 3.4.14), (–) h S i agrees with the composition Γ- E M -SSet ∗ (–)[ ω • ] −−−−→ Γ- I -SSet ∗ E ⊗ −−→ Spectra . Remark . Strictly speaking, Schwede works with the category
Spectra
Top of symmetric spectra in topological spaces , and he first passes to Γ- | E M| -Top ∗ via geometric realization. We omit the routine verification that after postcompos-ing with | – | : Spectra → Spectra
Top the above agrees with his construction; theonly non-trivial ingredient is the comparison between prolongations of Γ-spaces inthe simplicial and in the topological world, which appears for example as [
Sch18 ,Proposition B.29].We are now ready to define, see [
Sch19b , Definition 4.14]:
Definition . We write K gl for the composition ParSumCat ̥ −→ Γ- E M -Cat ∗ N −→ Γ- E M -SSet ∗ (–) h S i −−−−→ Spectra . If C is any parsummable category, then we call K gl ( C ) the global algebraic K -theory of C .The correct notion of ‘equivariant algebraic K -theory’ of symmetric monoidalcategories with G -action is not given by simply applying the non-equivariant Segal-Shimada-Shimakawa construction and pulling through the G -action; instead, thisrequired Shimakawa’s insight detailled in Example 2.2.11 above. Interestingly, itturns out that this is not necessary when generalizing from global to G -globalalgebraic K -theory: Definition . Let C be a G -parsummable category (i.e. a G -object in ParSumCat ). Its G -global algebraic K -theory K G -gl ( C ) is the G -spectrum ob-tained by equipping K gl with the G -action induced by functoriality. We write K G -gl : G -ParSumCat → G -Spectra for the induced functor.We will explain in the next subsection how this yields the G -global algebraic K -theory of a symmetric monoidal category with G -action, and prove (Theorem 4.1.28)that this indeed refines classical G -equivariant algebraic K -theory.We close this discussion by establishing a basic invariance property of G -global K -theory: .1. DEFINITION AND BASIC PROPERTIES 211 Definition . A map f : C → D of G -parsummable categories is called a G -global weak equivalence if the induced functor C ϕ → D ϕ is a weak equivalence(i.e. weak homotopy equivalence on nerves) for any universal subgroup H ⊂ M andany homomorphism ϕ : H → G .For G = 1, Schwede [ Sch19b , Definition 2.26] considered these under the name‘global equivalence.’ We can now prove the following generalization of [
Sch19b ,Theorem 4.16]:
Proposition . The functor K G -gl preserves G -global weak equivalences. For the proof it will be convenient to slightly reformulate the above constructionof K G -gl . This will use: Lemma . The nerve of a tame E M -category is tame again, and the in-duced functor N : E M - G -Cat τ → E M - G -SSet τ admits a preferred strong sym-metric monoidal structure. Proof.
See [
Len20a , Example 2.7 and Proposition 2.20]. (cid:3)
In particular, we get a canonical lift of N to a functor G -ParSumCat → G -ParSumSSet that we again denote by N. Corollary . The diagram G -ParSumCat Γ- E M - G -Cat ∗ G -ParSumSSet Γ- E M - G -SSet τ ∗ Γ- E M - G -SSet ∗ N ̥ N ̥ commutes up to canonical natural isomorphism. Proof.
The top arrow obviously factors through Γ- E M - G -Cat τ ∗ , so it suf-fices to construct a natural isomorphism filling G -ParSumCat Γ- E M - G -Cat τ ∗ G -ParSumSSet Γ- E M - G -SSet τ ∗ . N ̥ N ̥ But as in the proof of Theorem 2.3.1, it follows by abstract nonsense that the struc-ture isomorphisms of the strong symmetric monoidal functor N : E M - G -Cat τ → E M - G -SSet τ assemble into the desired isomorphism. (cid:3) Proof of Proposition 4.1.9.
By the previous corollary, we can factor K G -gl up to isomorphism as G -ParSumCat N −→ G -ParSumSSet ̥ −→ Γ- E M - G -SSet τ ∗ ֒ → Γ- E M - G -SSet ∗ (–)[ ω • ] −−−−→ Γ- G - I -SSet ∗ E ⊗ −−→ G -Spectra . Of these the first arrow preserves G -global weak equivalences as N preserves lim-its; moreover, the second arrow sends these to G -global level weak equivalencesby Proposition 2.3.16, which are preserved by the inclusion by definition and bythe penultimate arrow by Theorem 2.2.29. Finally, E ⊗ is homotopical by Corol-lary 3.4.20. (cid:3)
12 4. G -GLOBAL ALGEBRAIC K -THEORY G -global algebraic K -theory interacts very nicely with restrictions along grouphomomorphisms: Proposition . Let ϕ : H → G be any group homomorphism. Then thediagram G -ParSumCat G -Spectra ∞ G -global H -ParSumCat H -Spectra H -global( ϕ ∗ ) ∞ K ∞ G -gl ( ϕ ∗ ) ∞ K ∞ H -gl commutes up to canonical equivalence. Proof.
By construction, we even have an equality of homotopical functors ϕ ∗ ◦ K G -gl = K H -gl ◦ ϕ ∗ . (cid:3) G -global K -theory of symmetric monoidal G -categories. Recallthat a permutative category is a symmetric monoidal category in which the unitalityand associativity isomorphisms are required to be the respective identities. We write
PermCat for the 1-category of small permutative categories and strict symmetricmonoidal functors. In [
Sch19b , Constructions 11.1 and 11.6], Schwede constructsa specific functor Φ :
PermCat → ParSumCat as a variation of a ‘strictification’construction due to Schlichtkrull and Solberg [
SS16 , Sections 4.14 and 7]. We willbe able to completely blackbox the definition of Φ, and refer the curious reader toSchwede’s article instead.Unfortunately, the global algebraic K -theory of the parsummable categoryΦ( C ) is not yet the ‘correct’ definition of the global algebraic K -theory of C —forexample, if C is a small permutative replacement of the category of finite dimen-sional C -vector spaces and C -linear isomorphisms under ⊕ , then K gl (Φ C ) does notagree with the usual global algebraic K -theory of C , see [ Sch19b , Proposition 11.9]and in particular [
Len20b , Remark 3.27].4.1.2.1.
Saturation.
Ultimately, the above issue stems from the fact that whilewe have good control over the underlying category of Φ( C ) (which is equivalent to C itself), the equivariant information encoded in the E M -action is not that natural.On the other hand, there is an interesting class of parsummable G -categories forwhich the categorical information is enough to describe the underlying G -globalhomotopy type: Definition . A small E M - G category C is called saturated if the canon-ical map C ϕ ֒ → C ‘ h ’ ϕ = Fun( EH, ϕ ∗ C ) H from the honest fixed points to the cat-egorical homotopy fixed points is an equivalence of categories for each universalsubgroup H ⊂ M and each homomorphism ϕ : H → G .Here we use the notation ‘ h ’ instead of the usual h in order to emphasize thatwhile our notion of weak equivalences of G -parsummable categories is based onweak homotopy equivalences, the above are homotopy fixed points with respect tothe underlying equivalences of categories . In particular, (–) ‘ h ’ ϕ does not preserve G -global weak equivalences, i.e. ‘homotopy’ fixed points are not homotopy invariant. Lemma . Let f : C → D be a map of saturated G -parsummable categoriesand an equivalence of underlying categories. Then f is a G -global weak equivalence. Proof.
See [
Len20a , Lemma 1.28]. (cid:3) .1. DEFINITION AND BASIC PROPERTIES 213
While the parsummable G -categories Φ C are usually not saturated, our satu-ration construction , which appeared for G = 1 as [ Sch19b , Construction 7.18] andfor general G as [ Len20a , Construction 1.29], provides a universal way to remedythis situation:
Construction . Let C be a tame E M -category and equip Fun( E M , C )with the diagonal of the E M -action on C and the left E M -action induced by the right E M -action on E M via precomposition. We define C sat := Fun( E M , C ) τ and call it the saturation of C . The natural functor s : C → Fun( E M , C ) sendingan object c ∈ C to the constant functor at c restricts to C → C sat ; we omit the easyverification that this is natural with respect to the evident functoriality of (–) sat .Finally, we lift (–) sat to an endofunctor of E M - G -Cat τ by pulling throughthe G -action; we observe that s automatically defines a natural transformationid E M - G -Cat τ ⇒ (–) sat . Theorem . The functor (–) sat takes values in the saturated tame E M - G -categories. Moreover: (1) s : C → C sat is an underlying equivalence of categories for any C ∈ E M - G -Cat τ . (2) C sat ֒ → Fun( E M , C ) induces equivalences of categories on ϕ -fixed pointsfor every universal H ⊂ M and any ϕ : H → G ; in particular, it is a G -global weak equivalence. Proof.
This is [
Len20a , Theorem 1.30] and its proof. (cid:3)
As explained in [
Sch19b , Construction 7.23], the usual lax symmetric monoidalstructure on (–) sat with respect to the cartesian product restricts to a lax symmetricmonoidal structure with respect to the box product. In particular, (–) sat canonicallylifts to an endofunctor of G -ParSumCat . Using this we can now finally introduce: Definition . We define the G -global algebraic K -theory of permutative G -categories as the composition G -PermCat Φ −→ G -ParSumCat (–) sat −−−→ G -ParSumCat K G -gl −−−−→ G -Spectra , which we denote by K G -gl again. Remark . For G = 1 this recovers Schwede’s definition, see [ Sch19b ,discussion after Proposition 11.9].It is not hard to show that the above composition sends maps in G -PermCat that are equivalences of underlying categories to G -global weak equivalences. Wewill however be interested in the following more general notion: Definition . A G -equivariant functor f : C → D of small G -categoriesis called a G -global weak equivalence if the induced functor f ‘ h ’ ϕ : C ‘ h ’ ϕ → D ‘ h ’ ϕ is a weak homotopy equivalence for every finite group H and any homomorphism ϕ : H → G .For G = 1 this precisely the recovers the ‘global equivalences’ of [ Sch19a ]. Proposition . The functors (–) sat ◦ Φ : G -PermCat → G -ParSumCat and K G -gl : G -PermCat → G -Spectra preserve G -global weak equivalences. Proof.
The first statement is proven in [
Len20a , proof of Theorem 6.7]. Thesecond one now follows immediately from Proposition 4.1.9. (cid:3)
14 4. G -GLOBAL ALGEBRAIC K -THEORY An alternative description.
Let us write
SymMonCat for the cate-gory of small symmetric monoidal categories and strong symmetric monoidal cat-egories. As usual, we write G -SymMonCat for the corresponding category of G -objects; in particular, its objects are symmetric monoidal categories with a strict G -action through strong symmetric monoidal functors.We now want to extend the above construction of G -global algebraic K -theoryfrom G -PermCat to G -SymMonCat . A quick way to do this would be to useMac Lane’s strictification construction [ Mac98 , Theorem XI.3.1] to replace a givensmall symmetric monoidal G -category by an equivalent permutative category with G -action through strict symmetric monoidal functors; more precisely, the (non-full!) inclusion G -PermCat ֒ → G -SymMonCat is a homotopy equivalence withrespect to the underlying equivalences of categories, and hence in particular withrespect to the G -global weak equivalences, see [ Len20a , Remark 6.8].Here we will take a different route by giving a direct construction of the G -global algebraic K -theory of symmetric monoidal G -categories and then comparingit to the previous definition on G -PermCat . Definition . We write K ′ G -gl for the composition G -SymMonCat Γ −→ Γ- G -Cat ∗ Fun( E M , –) −−−−−−−→ Γ- E M - G -Cat ∗ N −→ Γ- E M - G -SSet ∗ (–) h S i −−−−→ G -Spectra . Here Γ is obtained from the Shimada-Shimakawa functor (see Example 2.2.5) bypulling through the G -action. Moreover, the left E M -action on Fun( E M , C ) isagain induced by the right E M -action on itself via precomposition.Our goal now is to prove: Theorem . The functor K ′ G -gl preserves G -global weak equivalences andthere exists a natural equivalence ( K ′ G -gl | G -PermCat ) ∞ ≃ K G -gl : G -PermCat ∞ → G -Spectra ∞ . The proof of the theorem relies on a comparison of the corresponding Γ- E M - G -categories. To this end we introduce: Definition . A map f in Γ- E M - G -Cat ∗ is called a G -global level weakequivalence if N( f ) is a G -global level weak equivalence in Γ- E M - G -SSet ∗ .We begin with the following simplification: Proposition . There exists a preferred zig-zag of levelwise G -global levelweak equivalences filling G -ParSumCat G -ParSumCatΓ- E M - G -Cat ∗ Γ- G -Cat ∗ Γ- E M - G -Cat ∗ . ̥ (–) sat ̥ forget Fun( E M , –) In particular, this yields a preferred equivalence between the corresponding functors G -ParSumCat → Γ- E M - G -Cat ∞∗ . Proof.
Firstly, we observe that the lax symmetric monoidal structure on (–) sat induces a natural transformation λ : ̥ ◦ (–) sat ⇒ (–) sat ◦ ̥ such that λ ( X )( n + ) is .1. DEFINITION AND BASIC PROPERTIES 215 given by the iterated structure map ( X sat ) ⊠ n → ( X ⊠ n ) sat , i.e. it fits by definitioninto a commutative diagram( X sat ) ⊠ n ( X ⊠ n ) sat ( X sat ) × n ( X × n ) sat λ ∼ = where the lower horizontal arrow is the canonical comparison map and the verticalarrows are (induced by) the inclusions. In particular, the lower horizontal arrow isan isomorphism (as the inclusion E M -Cat τ ֒ → E M -Cat preserves finite limitsand since both Fun( E M , –) and (–) τ are right adjoints), and the left hand verticalmap is a ( G × Σ n )-global weak equivalence by Corollary 2.1.14. On the otherhand, X ⊠ n ֒ → X × n is an underlying equivalence of categories [ Sch19b , proof ofTheorem 2.32], so the right hand vertical arrow is an underlying equivalence byTheorem 4.1.16 and hence a ( G × Σ n )-global weak equivalence by Lemma 4.1.14.Next, we observe that the inclusion C sat = Fun( E M , C ) τ ֒ → Fun( E M , C ) isa ( G × Σ S )-global weak equivalence for any tame E M -( G × Σ S )-category C byTheorem 4.1.16; applying this levelwise we conclude (–) sat ◦ ̥ ≃ Fun( E M , –) ◦ ̥ .Altogether, we are therefore reduced to showing that the composite Γ- E M - G -Cat ∗ forget −−−→ Γ- G -Cat ∗ triv E M −−−−−→ Γ- E M - G -Cat ∗ becomes equivalent to the identity after postcomposing both with the endofunctorFun( E M , –) of Γ- E M - G -Cat ∗ .But indeed, if C is any E M -( G × Σ S )-category, then we have a natural zig-zag C act ←−− E M × triv E M forget( C ) pr −→ triv E M forget( C )of underlying equivalences of categories, also see [ Len20a , proof of Lemma 6.10].The claim follows as Fun( E M , –) obviously sends underlying equivalences of cate-gories in E M -( G × Σ S )-Cat to ( G × Σ S )-global weak equivalences. (cid:3) Proof of Theorem 4.1.22.
We first show that the functors G -PermCat → G -Spectra ∞ are equivalent. By definition, it suffices for this to construct a naturallevelwise G -global level weak equivalence filling G -PermCat G -ParSumCat G -ParSumCatΓ- G -Cat ∗ Γ- E M - G -Cat ∗ . Γ Φ (–) sat ̥ Fun( E M , –) By the previous proposition, we are further reduced to constructing an equivalenceFun( E M , –) ◦ Γ ≃ Fun( E M , –) ◦ ̥ ◦ Φ . For this we fix an injection ϕ : { , } × ω ω . Then [ Sch19b , Construction 5.6]discusses a functor ϕ ∗ : ParSumCat → SymMonCat
16 4. G -GLOBAL ALGEBRAIC K -THEORY and it is mentioned without proof in [ Sch19b , Remark 11.4] that ϕ ∗ ◦ Φ is naturallyequivalent to the inclusion
PermCat ֒ → SymMonCat ; a full proof of this followsfrom [
Len20b , Proposition 2.15], also see [
Len20b , Remark 2.16].On the other hand, [
Sch19b , Construction 5.14] gives a natural transformation ̥ ⇒ Γ ◦ ϕ ∗ of functors ParSumCat → Γ-Cat , and this is a levelwise equivalenceof categories by [
Sch19b , proof of Theorem 5.15-(ii)].Altogether we therefore have a zig-zag of levelwise categorical equivalencesbetween Γ and ̥ ◦ Φ; as before, Fun( E M , –) sends this to G -global level weakequivalences, which completes the proof of the comparison.Finally, to prove that K ′ G -gl preserves G -global weak equivalences, we observethat the Γ-category Γ( C ) is special for any small symmetric monoidal category C (with G -action), so N ◦ Fun( E M , –) ◦ Γ( C ) is a special G -global Γ-space by the samearguments as before; it therefore suffices that Fun( E M , Γ( f )(1 + )) is a G -globalweak equivalence of E M - G -categories for any G -global weak equivalence f : C → D in G -SymMonCat . But, Γ( f ) agrees up to conjugation by isomorphisms with f , sothe claim follows immediately from the definitions as EH ֒ → E M is an equivalencein the 2-category of categories with right H -actions for any subgroup H ⊂ M . (cid:3) Remark . At first glance, the final comparison seems to depend on thechoice of the injection ϕ . However, as explained in [ Sch19b , Remark 5.8], the ϕ ∗ ’sfor varying ϕ assemble into a canonical functor E Inj( { , } × ω, ω ) × ParSumCat → SymMonCat , i.e. the individual functors are pairwise canonically isomorphic and these isomor-phisms are suitably coherent. We omit the straight-forward but slightly lengthyverification that the above equivalences ϕ ∗ ◦ Φ ≃ id and Γ ◦ ϕ ∗ ≃ ̥ are compatiblewith these comparison isomorphisms.Finally we note that we have analogously to Proposition 4.1.12: Proposition . Let ϕ : H → G be any group homomorphism. Then thediagram G -SymMonCat G -Spectra ∞ G -global H -SymMonCat H -Spectra H -global( ϕ ∗ ) ∞ ( K ′ G -gl ) ∞ ( ϕ ∗ ) ∞ ( K ′ H -gl ) ∞ commutes up to canonical equivalence. (cid:3) Comparison to G -equivariant algebraic K -theory. Let G be finite; theabove alternative description of the G -global algebraic K -theory of symmetricmonoidal categories with G -action allows for an easy comparison to G -equivariantalgebraic K -theory in the sense of Shimakawa: Definition . We write K G for the composition G -SymMonCat Γ −→ Γ- G -Cat ∗ Fun(
EG, –) −−−−−−−→ Γ- G -Cat ∗ N −→ Γ- G -SSet ∗ E G −−→ G -Spectra ;here G acts on EG from the right in the obvious way.If C is any small symmetric monoidal category with G -action, then K G ( C ) iscalled the G -equivariant algebraic K -theory of C . .1. DEFINITION AND BASIC PROPERTIES 217 Strictly speaking, Shimakawa uses a bar construction instead of the prolon-gation functor E G ; however, the two delooping constructions are equivalent by[ MMO17 , Sections 2–3].
Theorem . There are natural equivalences K ∞ G ≃ u G ◦ ( K G -gl ) ∞ : G -PermCat ∞ → G -Spectra ∞ G -equivariant and K ∞ G ≃ u G ◦ ( K ′ G -gl ) ∞ : G -SymMonCat ∞ → G -Spectra ∞ G -equivariant . Proof.
By Theorem 4.1.22 it suffices to prove the second statement. For thiswe let i : G → M be any injective homomorphism with universal image. Then i ∗ ω becomes a complete G -set universe, so Corollary 3.4.21 (for ϕ = id) shows that therectangle on the right in(4.1.1) Γ- E M - G -SSet ∗ Γ- G - I -SSet ∗ G -Spectra G -global Γ- E M - G -SSet ∗ Γ- G -SSet ∗ G -Spectra G -equivariant(–)[ ω • ]= E ⊗ ev ω ev i ∗ ω = i ∗ E G commutes up to a zig-zag of levelwise G -equivariant weak equivalences. On theother hand, the triangle on the left commutes up to equivalence by Theorem 2.2.29,reducing us to comparing the Γ- G -categories i ∗ Fun( E M , C ) and Fun( EG, i ∗ C ) forany Γ-category C . But Ei : EG → E M is an equivalence in the 2-category of right G -categories, so restricting along it produces the desired G -equivariant level weakequivalence. (cid:3) Remark . As before, the choice of i is actually inessential; for example,the right hand portion of (4 . .
1) can be parameterized over the contractible 2-groupoid with objects the injective homomorphisms G → M , 1-morphisms i → j the invertible ϕ ∈ M such that j ( g ) = ϕi ( g ) ϕ − for all g ∈ G (cf. Lemma 1.2.7),and a unique 2-cell between any pair of parallel arrows.Together with Proposition 4.1.26 (applied to the unique homomorphism G → K -theory and Shimakawa’s equivariant construction: Corollary . The diagram
SymMonCat Spectra ∞ global G -SymMonCat G -Spectra ∞ G -equivarianttriv ∞ K ∞ gl u G K ∞ G commutes up to natural equivalence. (cid:3)
18 4. G -GLOBAL ALGEBRAIC K -THEORY G -global algebraic K -theory as a quasi-localization In this final section, we will will use almost all of the theory developed above toprove, as the main results of this monograph, various G -global versions of Thoma-son’s classical equivalence between symmetric monoidal categories and connectivestable homotopy types.We begin with a version in the parsummable context: Theorem . The G -global algebraic K -theory functor K G -gl defines a quasi-localization G -ParSumCat → ( G -Spectra G -global ) ∞≥ . For G = 1, this in particular shows that global algebraic K -theory defines aquasi-localization K gl : ParSumCat → ( Spectra global ) ∞≥ , proving a conjecture ofSchwede [ Sch19b , p. 7] and one half of Theorem B from the introduction.The only missing ingredient for the proof of the above theorem is the followingcomparison, which we proved as [
Len20a , Theorem 5.8]:
Theorem . The nerve
N : G -ParSumCat → G -ParSumSSet descendsto an equivalence of the quasi-localizations at the G -global weak equivalences. (cid:3) Proof of Theorem 4.2.1.
At this point, this only amounts to collecting re-sults we proved above. We first recall from the proof of Proposition 4.1.9 that K G -gl agrees up to isomorphism with the composition G -ParSumCat N −→ G -ParSumSSet ̥ −→ Γ- E M - G -SSet τ ∗ ֒ → Γ- E M - G -SSet ∗ (–)[ ω • ] −−−−→ Γ- G - I -SSet ∗ E ⊗ −−→ G -Spectra ≥ ;moreover, if we equip these categories with the G -global weak equivalences or G -global level weak equivalences, then all of the above functors are homotopical.By the previous theorem, the first of these induces an equivalence on quasi-localizations; moreover, Theorem 2.1.46 implies that ̥ induces an equivalence G -ParSumSSet ∞ → ( Γ- E M - G -SSet τ, special ∗ ) ∞ , and by Corollary 2.2.49 thenext two functors also induce equivalences between the respective quasi-categoriesof special G -global Γ-spaces. Finally, Corollary 3.4.35 shows that E ⊗ induces a(Bousfield) localization ( Γ- G - I -SSet special ∗ ) ∞ → ( G -Spectra G -global ) ∞≥ . (cid:3) In fact, the above argument also shows:
Theorem . The functor G -ParSumCat → Γ- G - I -SSet ∗ from the def-inition of K G -gl descends to an equivalence between the quasi-categories of G -par-summable categories (with respect to the G -global weak equivalences) and the special G -global Γ -spaces (with respect to the G -global level weak equivalences). (cid:3) We can view this as a ‘non-group-completed’ version of the G -global ThomasonTheorem in the spirit of Mandell [ Man10 , Theorem 1.4].
Remark . In view of the other comparisons established in Chapter 2,this also yields equivalences between G -parsummable categories on the one handand G -ultra-commutative monoids or any of the other notions of special G -globalΓ-spaces discussed above on the other hand.Next, we will establish the corresponding results for the G -global algebraic K -theory of permutative (or symmetric monoidal) categories with G -action. For thiswe will need the following comparison which we proved as [ Len20a , Theorem 6.7]: .2. G -GLOBAL ALGEBRAIC K -THEORY AS A QUASI-LOCALIZATION 219 Theorem . The composition G -PermCat Φ −→ G -ParSumCat (–) sat −−−→ G -ParSumCat descends to an equivalence of the quasi-localizations at the G -global weak equiva-lences. We now immediately get Theorem A from the introduction, which in particularsubsumes the remaining half of Theorem B:
Theorem . The functors K G -gl : G -PermCat → ( G -Spectra G -global ) ∞≥ and K ′ G -gl : G -SymMonCat → ( G -Spectra G -global ) ∞≥ are quasi-localizations. Proof.
The first statement follows from the previous theorem together withTheorem 4.2.1. Moreover, as G -PermCat ֒ → G -SymMonCat is a homotopyequivalence with respect to the G -global weak equivalences, the second statementnow follows from Theorem 4.1.22. (cid:3) Theorem . The composition G -SymMonCat Γ −→ Γ- G -Cat ∗ Fun( E M , –) −−−−−−−→ Γ- E M - G -Cat ∗ N −→ Γ- E M - G -SSet ∗ defines an equivalence G -SymMonCat ∞ G -gl. ≃ ( Γ- E M - G -SSet special ∗ ) ∞ G -gl. level . Proof.
Again it suffices to prove this after restricting to G -PermCat . As wehave seen in the proof of Theorem 4.1.22, the resulting functor is equivalent to thecomposite G -PermCat (–) sat ◦ Φ −−−−−→ G -ParSumCat → Γ- E M - G -SSet ∗ where the second functor is the composite discussed before. The claim thereforefollows from Theorem 4.2.5 together with the proof of Theorem 4.2.1. (cid:3) Again, this in particular yields the corresponding statements for Schwede’sglobal algebraic K -theory. We close this discussion by establishing the correspond-ing results for G -equivariant algebraic K -theory: Theorem . The composition G -SymMonCat Γ −→ Γ- G -Cat ∗ Fun(
EG, –) −−−−−−−→ Γ- G -Cat ∗ N −→ Γ- G -SSet ∗ yields a quasi-localization G -SymMonCat → ( Γ- G -SSet special ∗ ) ∞ G -equiv. level . Proof.
By the proof of Theorem 4.1.28, the above agrees up to G -equivariantlevel weak equivalence with the composition G -SymMonCat → Γ- E M - G -SSet ∗ (–)[ ω • ] −−−−→ Γ- G - I -SSet ∗ ev U −−→ Γ- G -SSet ∗ where the unlabelled arrow is the composite from the previous theorem, and U isany complete G -set universe. The claim now follows from the previous theoremtogether with Corollary 2.2.49 and Theorem 2.2.55. (cid:3) Remark . We emphasize that we are working with the na¨ıve notion ofsymmetric monoidal or permutative categories with G -action here, not with the genuine permutative G -categories in the sense of Guillou and May [ GM17 , Defi-nition 4.5]. Following Shimakawa, they observed in Proposition 4.6 of op.cit. thatthe endofunctor Fun(
EG, –) of G -PermCat lifts to a functor to the category ofgenuine permutative G -categories, and they further mention that (while expecting
20 4. G -GLOBAL ALGEBRAIC K -THEORY other examples to exist) they are not aware of any genuine permutative G -categorynot arising this way.Our above theorem actually suggests that their construction should yield anequivalence of homotopy theories between G -PermCat (with respect to the mapsinducing weak equivalences on ‘homotopy’ fixed points) and the genuine permuta-tive G -categories (with respect to maps inducing weak equivalences on honest fixedpoints); in particular, up to G -equivariant weak equivalence any genuine permuta-tive G -category should indeed arise this way. We plan to come back to this questionin future work.Finally, we can now prove Theorem C from the introduction: Theorem . The equivariant K -theory functor K G : G -SymMonCat → ( G -Spectra G -equivariant ) ∞≥ is a quasi-localization. Proof.
This follows from the previous theorem by the usual G -equivariantDelooping Theorem (recalled as Theorem 3.4.10 above). (cid:3) PPENDIX A
Abstract homotopy theory
In this appendix we collect for easy reference some general results from theliterature about quasi-categories and model categories that are used over and overin the main text.
A.1. Quasi-localizations
While we in most cases use tools from homotopical algebra to prove our state-ments, we are ultimately interested in comparisons on the level of quasi-categories.The passage from the former to the latter is (as usual) provided by the followingdefinition:
Definition
A.1.1 . Let C be a quasi-category and let W ⊂ C be a collection ofmorphisms. A functor γ : C → D of quasi-categories is called a quasi-localization at W if it has the following universal property: for every quasi-category T , restrictionalong γ induces an equivalence(A.1.1) Fun( D , T ) → Fun W ( C , T ) , where Fun W ( C , T ) ⊂ Fun( C , T ) is the full subcategory spanned by those functorsthat send morphisms in W to equivalences. (We remark that taking T = D impliesthat γ itself sends morphisms in W to equivalences.)By common abuse of notation, we will often supress W from notation andcall γ : C → D (and in fact, by further abuse also simply D itself) ‘the’ quasi-localization of C . Warning
A.1.2 . Lurie [
Lur09 , Definition 5.2.7.2] uses the term ‘localization’for a functor with a fully faithful right adjoint, for which we use the more classi-cal name ‘Bousfield localization’ in this paper; the above terminology is used forexample by Joyal in [
Joy08 ].While every Bousfield localization is a quasi-localization by [
Lur09 , Proposi-tion 5.2.7.12], the converse does not hold and in fact most of the quasi-localizationswe are interested in are not Bousfield localizations.
Remark
A.1.3 . In the 1-categorical situation one often adds the conditionthat W is a wide subcategory (in which case the pair ( C , W ) is called a relativecategory ) and sometimes also that it satisfies the 2-out-of-3 property ( categorieswith weak equivalences ) or even 2-out-of-6 ( homotopical categories ). While in all ofour applications W is indeed a wide subcategory, the stronger properties need notalways be satisfied.However, assume γ : C → D is a quasi-localization at some collection W .Then by definition any functor sending W to equivalences factors up to equivalencethrough γ and hence it sends (by 2-out-of-3 for equivalences in quasi-categories)more generally all morphisms f ∈ W to equivalences where W is the collection of morphisms sent to equivalences by γ . It follows that γ is also a quasi-localizationat any collection W ′ of morphisms such that W ⊂ W ′ ⊂ W . As W is a widesubcategory satisfying the 2-out-of-6 property, this will allow us in some proofs torestrict to this case. Remark
A.1.4 . Specializing T in ( A. .
1) to nerves of categories and using the(enriched) adjunction h ⊣ N, we see that if C → D is a quasi-localization at somecollection W of morphisms, then the induced functor h C → h D is a localizationat the same collection. In particular, if C is an ordinary relative category andN C → D is a quasi-localization, then C ∼ = hN C → h D is an ordinary localization,where the isomorphism on the left is the inverse of the counit. A.1.1. Simplicial localization.
While we use the above theory mostly as aconceptual way to pass from homotopical algebra to higher category theory, thereis also a simplicial version of the above (which historically predates the systematicstudy of higher categories), that is central to some arguments in Chapter 1.For this we recall that a simplicial category C can be equivalently viewed asa simplicial object in categories that is constant on objects. We will write C n (which is an ordinary category) for the n -simplices of the corresponding simplicialobject. For n = 0 this yields what is usually called the underlying category of C and accordingly we will write u C := C . Definition
A.1.5 . Let C be a simplicial category that is fibrant in the Bergnermodel structure [ Ber07 ], i.e. all its mapping spaces are Kan complexes. Moreover,let W be any collection of morphisms in u C . A simplicially enriched functor γ : C → D is called a simplicial localization at W if for some (hence any) fibrantreplacement D → E the induced map N ∆ C → N ∆ E is a quasi-localization at W (considered as a collection of morphisms in N ∆ ( C ) in the obvious way). Construction
A.1.6 . We refer the reader to [
DK80a , 2.1] for the definitionof the
Hammock localization L H of a relative category ( C , W ). We will not needany details about the construction (except in the proof of Proposition A.1.15, wherethe necessary properties will be recalled), but only that it is a strict 1-functor andthat there is a natural map C → L H ( C , W ).We also recall from [ DK80a , Remark 2.5] that this definition can be extendedto a simplicial category C together with a (wide) simplicial subcategory W as fol-lows: we define L H ( C , W ) to be the diagonal of the bisimplicial object in categoriesobtained by applying L H levelwise to ( C n , W n ). We remark that L H becomesa functor by employing functoriality of L H levelwise, and we get a natural map C → L H ( C , W ) induced from the unenriched situation. Definition
A.1.7 . A relative simplicial category consists of a simplicial cate-gory C together with a wide simplicial subcategory W . We call the pair ( C , W ) fibrant if both C and W are fibrant in the Bergner model structure. Theorem
A.1.8 (Dwyer & Kan, Hinich) . Let ( C , W ) be a fibrant relative sim-plicial category. Then the natural map C → L H ( C , W ) is a simplicial localizationat the -morphisms of W . Proof.
This is [
Hin16 , Proposition 1.2.1], also cf. [
Hin16 , 1.1.3]. (cid:3)
Note that this theorem in particular tells us that if ( C , W ) is an ordinaryrelative category, then C → L H ( C , W ) is a simplicial localization. .1. QUASI-LOCALIZATIONS 223 Remark
A.1.9 . Let C be a relative category. Then the canonical map C → L H ( C ) is the identity on objects. The standard fibrant replacement in the Bergnermodel structure proceeds by applying Kan’s Ex ∞ -functor to morphism spaces andhence again is the identity on objects, i.e. we get a simplicial localization γ : C → D with D fibrant such that γ is the identity on objects.As the objects of N C respectively N ∆ ( D ) are canonically identified with theobjects of C respectively D , we conclude that any relative category C admits aquasi-localization N( C ) → E that is an isomorphism on objects. In this case wecan of course just rename the 0-simplices of E so that our quasi-localization isthe identity on objects. We will call the quasi-localization constructed this way(and by abuse of terminology, also the resulting quasi-category E ) the associatedquasi-category of C . We write E =: C ∞ .Of course, insisting on a statement about equality (or already about isomor-phism) of the class of objects is ‘evil’ in the sense that it is not invariant underequivalences. However, we think that this convention allows us to simplify somestatements and it is moreover also close to the way we usually think about quasi-localizations (as higher versions of the ordinary homotopy category, whose standardconstruction is the identity on objects).Dwyer and Kan already proved that for a simplicial model category C its fullsubcategory C ◦ of cofibrant-fibrant objects is the simplicial localization of u C ◦ at the weak homotopy equivalences; more precisely, the inclusion u C ◦ ֒ → C ◦ is aquasi-localization, see [ DK80b , Proposition 4.8]. Their proof in fact gives a generalcriterion, which we exploit in the main text:
Proposition
A.1.10 . Let C be a fibrant simplicial category and let W ⊂ u C be a wide subcategory all of whose morphisms are homotopy equivalences in C .Assume moreover that for every n ≥ the map L H ( C , W ) → L H ( C n , s ∗ W ) induced from the unique map s : [ n ] → [0] is a Dwyer-Kan equivalence (i.e. it inducesan equivalence of Ho(
SSet ) -enriched homotopy categories). Then u C ֒ → C is asimplicial localization at W . Proof.
This is implicit in [
DK80b , Proof of Proposition 4.8], also cf. [
Hin16 ,1.4.3 and 1.4.4]. We begin with the following observation:
Claim.
The canonical map C → L H ( C , W ) is a Dwyer-Kan equivalence. Proof.
Let L H ( C , W ) → E be a fibrant replacement. Then Theorem A.1.8implies that the induced map N ∆ ( C ) → N ∆ ( E ) is a quasi-localization at the equiv-alences of C , hence it is an equivalence of quasi-categories. As N ∆ reflects weakequivalences between fibrant simplicial categories (as the right half of a Quillenequivalence), we conclude that C → E is a Dywer-Kan equivalence. The claimfollows from 2-out-of-3. △ Looking at the naturality squareu
C C L H (u C , W ) L H ( C , W )
24 A. ABSTRACT HOMOTOPY THEORY it therefore suffices by 2-out-of-3 that L H (u C , W ) → L H ( C , W ) is a Dwyer-Kanequivalence. This map is the identity of objects so it suffices to prove that it isgiven on mapping spaces by weak equivalences of simplicial sets.For this we observe that for objects X, Y ∈ C the mapping space on the rightis by definition the diagonal of the bisimplicial set maps L H ( C • ,W ) ( X, Y ), and themap maps L H ( C ,W ) ( X, Y ) → maps L H ( C • ,W ) ( X, Y ) in question is the diagonal of themap of bisimplicial setsconst maps L H ( C ,W ) ( X, Y ) → maps L H ( C • ,W ) ( X, Y )induced in degree n by the degeneracy [ n ] → [0]. The assumption guarantees thatthis is a levelwise weak equivalence and hence its diagonal is a weak equivalence bythe ‘Diagonal Lemma’ from simplicial homotopy theory, see [ GJ99 , Theorem 4.1.9],finishing the proof. (cid:3)
Remark
A.1.11 . The same proof works in the case that W is any wide fibrantsimplicial subcategory (in which one has to replace s ∗ W by W n ). However, we willonly need the above version.We recall one of the standard ways to produce Dwyer-Kan equivalences onsimplicial localizations, also see [ DK80b , 2.5].
Definition
A.1.12 . Let
F, G : C → D be homotopical simplicial functors ofrelative simplicial categories. Then F and G are called homotopic if they can beconnected by a zig-zag of simplicially enriched transformations that are at the sametime levelwise weak equivalences. Definition
A.1.13 . A homotopical functor F : C → D of relative simpliciallyenriched categories is called a homotopy equivalence if there exists a homotopicalfunctor G : D → C such that F G is homotopic to the identity of D and GF ishomotopic to the identity of C . Corollary
A.1.14 . Let F : C → D be a homotopy equivalence of fibrant sim-plicial categories equipped with fibrant wide subcategories of weak equivalences. Thenthe induced functor on L H is a Dwyer-Kan equivalence. Proof.
This follows from the universal property in the same way as in theclaim in the proof of Proposition A.1.10.Alternatively, we reduce to the case of ordinary categories as in the proof ofProposition A.1.10. We can then enlarge the weak equivalences on both sides sothat they satisfy 2-out-of-3, so that all the intermediate functors in the zig-zags arethemselves homotopical. The claim then follows by inductively applying [
DK80a ,Proposition 3.5]. (cid:3)
Finally we note:
Proposition
A.1.15 . Let C be a model category with functorial factorizationsand let B ⊂ C be a full subcategory closed under weak equivalences. Then theinclusion B ֒ → C induces a fully faithful functor on quasi-localizations. Proof.
It suffices that L H ( B ) → L H ( C ) is a weak equivalence on morphismspaces. For this let us pick X, Y ∈ B arbitrary.We consider the category M X,Y with objects the zig-zags
X A B Y ∼ ∼ .1. QUASI-LOCALIZATIONS 225 and morphisms the commutative diagrams
A BX Y.A ′ B ∼ ∼ ∼ ∼∼∼ The important observation is that it does not matter whether we form this in B or C as B is closed under weak equivalences. In the terminology of [ DK80a ],the k -simplices of M X,Y are hammocks of width k and length 3 (in B or equiv-alently in C ) between X and Y , whereas the k -simplices of maps L H ( B ) ( X, Y )resp. maps L H ( C ) ( X, Y ) are reduced hammocks of width k and arbitrary lengthbetween X and Y in B resp. C . We then have a commutative diagramN( M X,Y ) maps L H ( B ) ( X, Y )N( M X,Y ) maps L H ( C ) ( X, Y )where the horizontal maps are given by reduction (i.e. iteratively removing identitycolumns and composing adjacent horizontal arrows in the same direction). Thelower map is a weak homotopy equivalence by [
DK80b , 7.2]; it therefore sufficesthat the top map is also a weak equivalence, for which we can use the same argumentas them. Namely, it is enough by [
DK80a , Proposition 6.2-(i)] together with[
DK80a , Proposition 8.2] to exhibit subcategories W , W of the weak homotopyequivalences in B satisfying the following conditions:(1) Any diagram X YX ′ i ∈ W with i ∈ W can be functorially completed to a square X YX ′ Y ′ i ∈ W j ∈ W with j ∈ W .(2) Any diagram Y ′ X Y p ∈ W with p ∈ W can be functorially completed to a square X ′ Y ′ X Y q ∈ W p ∈ W with q ∈ W .
26 A. ABSTRACT HOMOTOPY THEORY (3) Every weak equivalence w admits a functorial factorization w = w w with w i ∈ W i , i = 1 , B is closedunder weak equivalences together with the respective properties for C , cf. [ DK80a ,Proposition 8.4], but as Dwyer and Kan omit most of the (simple) argument forthese, we give a full proof here for completeness:We take W to be the acyclic cofibrations and W to be the acyclic fibrations.Then invoking the functorial factorizations of C in, say, an acyclic cofibration fol-lowed by a fibration (which is automatically acyclic by 2-out-of-3) we get for anyweak equivalence w : X → Y in B a functorial diagram X w ∈ W −−−−−→ H w ∈ W −−−−−→ Y in C ; as B is closed under weak equivalences, also H ∈ B , and as B is full, thisyields the desired functorial factorizations, proving Condition (3).It remains to verify Condition (1); Condition (2) then follows from duality.For this we simply pass to pushouts in C ; then j is again an acylic cofibration byabstract nonsense about model categories and hence also Y ′ ∈ B . Functoriality isinduced (and in fact, uniquely determined) by the universal property. This finishesthe proof. (cid:3) A.1.2. Quasi-localizations and categories of diagrams.
Let A and B besimplicial categories; we write A B or alternatively Fun ( B , A ) for the simpliciallyenriched functor category.In this subsection, we want to prove the following model categorical manifes-tation of the universal property of quasi-localizations: Theorem
A.1.16 (Dwyer & Kan) . Let
A, B be small and fibrant simplicialcategories, let f : A → B be a simplicial localization at some collection W of mor-phisms, and let C be any combinatorial simplicial model category. Then the functorinduced by f ∗ : C B → C A on associated quasi-categories is fully faithful and its es-sential image consists precisely of those elements of C A that send morphisms in W to weak homotopy equivalences in C . The fibrancy assumptions on A and B are not necessary, but merely an artifactof our methods. In the special case that C = SSet with the usual Kan-Quillenmodel structure (which is the only case we use in the main text), a proof withoutthis assumption can be found as [
DK87 , Theorem 2.2]. However, the language andsetup used in op.cit. differ from ours, and rigorously translating their statementwould require quite a bit of additional terminology.Instead, we will therefore give an alternative proof as a consequence of a ‘rigid-ification’ result due to Lurie. This requires some preparations, throughout whichwe fix the combinatorial simplicial model category C . Lemma
A.1.17 . Let A be a small simplicial category and equip C A with eitherthe projective or injective model structure. Then each of the inclusions ( C A ) ◦ ֒ → ( C ◦ ) A ֒ → C A induces an equivalence on quasi-localizations of underlying 1-categories at the lev-elwise weak equivalences. .1. QUASI-LOCALIZATIONS 227 Proof.
This is true for the composition (in even greater generality) by [
Hin16 ,Proposition 1.3.8]; we remark that in our case a simpler proof can be given analo-gously to [
DK80b , 7.1]. We will use the same strategy to prove that the first mapalso has the desired property, which is enough to prove the lemma.The model category C A is combinatorial and hence we can find in particularfunctorial cofibrant and fibrant replacements, i.e. a functor Q : C A → C A togetherwith a natural transformation π : Q ⇒ id and a functor P : C A → C A together witha natural transformation ι : id ⇒ P such that for each X ∈ C A , π X : QX → X is anacyclic fibration with cofibrant source and ι X : X → P X is an acyclic cofibrationwith fibrant target; we emphasize that we do not make any claim about thesefunctors and transformations being simplicially enriched. We observe that P and Q are homotopical, preserve ( C A ) ◦ and ( C ◦ ) A (the latter because (co)fibrations in C A are in particular levelwise (co)fibrations), and that moreover P Q sends all of C A to ( C A ) ◦ .We claim that (the restriction of) P Q is homotopy inverse to the inclusionu (cid:0) ( C A ) ◦ (cid:1) ֒ → u (cid:0) ( C ◦ ) A (cid:1) . Indeed, we have for each X ∈ C A a natural zig-zag oflevelwise weak equivalences P Q ( X ) ( ι Q ◦ X ) ←−−−− Q ( X ) π X −−→ X in C A . By the above remarks, Q preserves both ( C ◦ ) A (hence the above exhibits P Q as right homotopy inverse to the inclusion) and ( C A ) ◦ (hence the above exhibits P Q also as left homotopy inverse); this finishes the proof. (cid:3)
Proposition
A.1.18 . Let A be a small fibrant simplicial category. Then thecomposition N (cid:0) ( C ◦ ) A (cid:1) ֒ → N ∆ (cid:0) ( C ◦ ) A (cid:1) → N ∆ ( C ◦ ) N ∆ ( A ) , where the second map is adjunct to N ∆ (cid:0) ( C ◦ ) A (cid:1) × N ∆ ( A ) ∼ = N ∆ (cid:0) ( C ◦ ) A × A (cid:1) eval −−→ N ∆ ( C ◦ ) , is a quasi-localization at the levelwise weak equivalences. Proof.
Equip C A with either the projective or injective model structure. Bythe previous lemma we may restrict to ( C A ) ◦ , and this can then be factored asN (cid:0) ( C A ) ◦ (cid:1) ֒ → N ∆ (cid:0) ( C A ) ◦ (cid:1) ֒ → N ∆ (cid:0) ( C ◦ ) A (cid:1) → N ∆ ( C ◦ ) N ∆ ( A ) . Here the first map is a quasi-localization at the levelwise weak equivalences by[
DK80b , Proposition 4.8], while the composition of the latter two maps is anequivalence by a special case of [
Lur09 , Proposition 4.2.4.4], also see [
Lur09 , proofof Corollary 4.2.4.7]. Thus, the whole composition is a quasi-localization at thelevelwise weak equivalences, finishing the proof. (cid:3)
Proof of Theorem A.1.16.
Let us say that the essential image of somefunctor is created by a collection X if it consists precisely of objects equivalentto X . The statement of the proposition is then equivalent to demanding that theessential image of f ∗ be created by those diagrams that send morphisms in W toweak homotopy equivalences (as this collection is itself closed under equivalences,by the 2-out-of-3-property for weak equivalences in C and since weak equivalencesin C A are saturated). This notion has the advantage that it is invariant underequivalences.
28 A. ABSTRACT HOMOTOPY THEORY
We observe that the subcollection of levelwise cofibrant-fibrant diagrams in-verting W creates the same essential image: indeed, by the proof of the previousproposition, any X ∈ C A admits a zig-zag of levelwise weak equivalences to onethat is levelwise cofibrant-fibrant, and if X inverts W , so does this replacement by2-out-of-3.Now let us consider the diagramN (cid:0) ( C ◦ ) B (cid:1) N( C B ) D N (cid:0) ( C ◦ ) A (cid:1) N( C A ) E f ∗ f ∗ where the right hand horizontal maps are quasi-localizations and the right handvertical map is induced by f ∗ ; the right hand square commutes up to equivalencewhile the left hand one commutes strictly. By Lemma A.1.17 also the horizontalcomposites are quasi-localizations at the levelwise weak equivalences and obviouslythe right hand vertical map still qualifies as induced map. Altogether we see that itsuffices to prove: the functor induced by f ∗ : ( C ◦ ) B → ( C ◦ ) A on quasi-localizationsis fully faithful and its essential image is generated by those elements of ( C ◦ ) A thatinvert W (or, more precisely, their images under the quasi-localization functor).For this we may pick any model of quasi-localizations, and we choose the onefrom the previous proposition. We then have a (strictly) commutative diagramN (cid:0) ( C ◦ ) B (cid:1) N ∆ ( C ◦ ) N ∆ ( B ) N (cid:0) ( C ◦ ) A (cid:1) N ∆ ( C ◦ ) N ∆ ( A ) f ∗ N ∆ ( f ) ∗ which allows us to identify the induced map with the obvious restriction. Byassumption, this is fully faithful with essential image those N ∆ ( A ) → N ∆ ( C ◦ ) thatinvert W , which obviously includes all images of diagrams in ( C ◦ ) A sending W to weak homotopy equivalences (as weak homotopy equivalences in C ◦ agree withhonest homotopy equivalences). So it only remains to prove the following converse:each such diagram N ∆ ( A ) → N ∆ ( C ◦ ) is equivalent to the image of an object of C A inverting W . But indeed, as quasi-localizations are essentially surjective (seeRemark A.1.9), it is equivalent to some X ∈ ( C ◦ ) A , which by another applicationof 2-out-of-3 sends W to (weak) homotopy equivalences as desired. (cid:3) A.1.3. Derived functors.
Assume we are given a Quillen adjunction(A.1.2) F : C ⇄ D : G. While the above associates quasi-categories to C and D , in most cases F and G are not homotopical, so they a priori do not give rise to functors between them. Inthe classical situation, one instead has left and right derived functors , respectively,yielding an adjunction L F : Ho( C ) ⇄ Ho( D ) : R G .We are interested in the following higher categorical version of this: Theorem
A.1.19 (Mazel-Gee) . In the above situation there is an adjunction (A.1.3) L F : C ∞ ⇄ D ∞ : R G .2. SOME HOMOTOPICAL ALGEBRA 229 of associated quasi-categories that is induced by ( A. . in the following sense: therestriction of L F to C c is equivalent (in a preferred way) to the composition C c F −→ D → D ∞ , and dually for the restriction of R G to D f . Proof.
This is [
Maz16 , Theorem 2.1] and its proof. (cid:3)
In analogy with the classical situation we call L F the left derived functor of F and R G the right derived functor of G . The adjunction ( A. .
3) is called the derivedadjunction . Remark
A.1.20 . By [
DK80b , Proposition 5.2] the inclusions C c ֒ → C and D f ֒ → D induce equivalences on quasi-localizations, also see [ Maz16 , Lemma 2.8]or [
Hin16 , Proposition 1.3.4]. It follows that the above functors L F and R G are determined up to (canonical) equivalence by their restrictions to C c or D f ,respectively, prescribed above.Now assume F is in fact homotopical. Then it induces F ∞ : C ∞ → D ∞ bythe universal property of quasi-localizations. But the restriction of F ∞ to C c is bydefinition equivalent to the restriction of L F , so we conclude from the above that L F ≃ F ∞ in this case. Dually, we see that R G ≃ G ∞ whenever G is homotopical. Remark
A.1.21 . The above characterization together with Remark A.1.4 al-ready implies that we have (canonical) identifications of the functors induced by L F and R G on the homotopy categories with the classical derived functors.However, it is a bit subtle to identify the unit or counit transformation of thequasi-categorical adjunction ( A. . C and D admit functorialfactorizations, there are obvious candidates of this, mimicking the construction ofthe derived adjunction on homotopy categories, and Mazel-Gee sketches a proofthat one can use this to get a unit transformation in [ Maz16 , A.3.1].While all the model categories appearing in this monograph admit functorialfunctorizations, we do not need this identification. In fact, we only care aboutthe adjunction data to exist and not how they actually look like: namely, we onlywant to prove that certain functors between quasi-categories are fully faithful oreven equivalences, which for adjunctions can be checked on the level of homotopycategories.This homotopy categorical statement can then (by the above identification)simply be checked for the standard construction of the adjunction induced on ho-motopy categories by a Quillen adjunction, for which the usual unit and counit canbe used as a tool to establish full faithfulness etc. In particular we see that ( A. . F ⊣ G is a Quillen equivalence.We emphasize that this argument does not need any information about theactual isomorphism between the classical left derived functor of F and the functorinduced by L F on homotopy categories (or dually for right derived functors of G ). A.2. Some homotopical algebraA.2.1. Bousfield localizations of model categories.
Let C be any modelcategory. We recall that a (left) Bousfield localization of C is a model structure C loc on the same underlying category with the same cofibrations as C and suchthat each weak equivalence of C is also a weak equivalence in C loc . If W loc is the
30 A. ABSTRACT HOMOTOPY THEORY collection of weak equivalences in C loc , we also call C loc the Bousfield localizationof C with respect to W loc .It follows directly from the definitions that(A.2.1) id : C ⇄ C loc : idis a Quillen adjunction with homotopical left adjoint. To see that ( A. .
1) inducesa Bousfield localization on associated quasi-categories, we have to show that R id isfully faithful, which one can either prove directly from the usual construction of thederived counit or instead by abstract nonsense as any adjoint of a quasi-localizationis fully faithful [ Cis19 , Proposition 7.1.17].We will now discuss several criteria for the existence of Bousfield localizationsof model categories.A.2.1.1.
Bousfield’s criterion.
The first such existence criterion appeared as[
BF78 , Theorem A.7], and it was later simplified by Bousfield to the followingstatement:
Theorem
A.2.1 (Bousfield) . Let C be a right proper model category, let Q : C → C be a functor and let η : id ⇒ Q be a natural transformation. We call a map f in C a Q -weak equivalence if Qf is a weak equivalence.Assume the following conditions are satisfied: (1) Q is homotopical. (2) For each X ∈ C , both η QX and Qη X are weak equivalences. (3) If A BC D g pf is any pullback square in C such that f is a Q -weak equivalence, p is afibration, B, D are fibrant and such that moreover η B , η D are weak equiv-alences, then g is a Q -weak equivalence.Then the Bousfield localization C loc of C at the Q -weak equivalences exists. More-over, a map f : X → Y is a fibration in C loc if and only if it is a fibration in C andthe square X QXY QY f η Qfη is a homotopy pullback in C . An object X is fibrant in C loc , if and only if it isfibrant in C and η X : X → QX is a weak equivalence in C .Finally, C loc is left proper provided that C is. Proof. [ Bou01 , Theorem 9.3 and Remark 9.5] proves all of this except forthe characterization of fibrant objects, which in turn is [
Sta08 , Lemma 2.5-(1)]. (cid:3)
Remark
A.2.2 . [ Sta08 , Theorem 1.1] also shows that we can drop the con-dition that C be right proper in the above theorem (at the cost of losing thecharacterization of the fibrations). .2. SOME HOMOTOPICAL ALGEBRA 231 A.2.1.2.
Set theoretic criteria.
If one is willing to assume local presentabilityof the base model category C , one can employ set theoretic techniques in order toconstruct Bousfield localizations in great generality. This is based on the followingnotions: Definition
A.2.3 . Let C be a simplicial model category, and write h X, Y i forthe mapping spaces in the Ho( SSet )-enriched homotopy category of C ; explicitly, h X, Y i can be computed as the mapping space in C between a cofibrant replacementof X and a fibrant replacement of Y .Now let S be any set of maps in C .(1) An object Z ∈ C is called S -local if h f, Z i is an isomorphism in Ho( SSet )for all f ∈ S .(2) A map g : X → Y in C is called an S -weak equivalence if h f, Z i is anisomorphism in Ho( SSet ) for all S -local Z ∈ C .We remark that in the second condition we could equivalently ask that theinduced map [ f, Z ] of hom-sets in the unenriched homotopy category be bijective(as the S -local objects are closed under the cotensoring over SSet ). However, inthe definition of S -locality the use of the mapping spaces is indeed essential. Theorem
A.2.4 . Let C be a left proper, combinatorial, and simplicial modelcategory, and let S be a set of maps in C . Then there exists a (unique) modelstructure C loc on C with the same cofibrations as C and the S -weak equivalencesas weak equivalences. This model structure is left proper, simplicial, and combina-torial. Moreover, its fibrant objects consist precisely of the fibrant objects of C thatare in addition S -local. Proof.
The notion of S -locality is obviously invariant under changing the el-ements of S by conjugation with weak equivalences, and hence so is the notionof S -weak equivalences. We may therefore assume without loss of generality that S consists of cofibrations of C , for which the above appears as [ Lur09 , Proposi-tion A.3.7.3]. (cid:3)
Theorem
A.2.5 (Lurie) . Let (A.2.2) F : C ⇄ D : G be a simplicial Quillen adjunction of left proper simplicial combinatorial model cate-gories. Assume that the right derived functor R G is fully faithful (say, as a functorbetween homotopy categories). Then there exists a Bousfield localization C loc of C whose weak equivalences are precisely those maps f such that ( L F )( f ) is an isomor-phism in Ho( D ) . This model structure is again left proper and an object is fibrantin C loc if and only if it is fibrant in C and contained in the essential image of R G .Moreover, ( A. . defines a simplicial Quillen equivalence (A.2.3) F : C loc ⇄ D : G. Proof. [ Lur09 , Corollary A.3.7.10] proves all of this except the characteriza-tion of the fibrant objects. We record that the proof of the aforementioned corollaryuses [
Lur09 , Proposition A.3.7.3] (i.e. the previous theorem) to construct the modelstructure, so the fibrant objects of C loc are closed inside the fibrant objects of C under the weak equivalences of C (this is in fact true for any Bousfield localization,but I do not know a good reference for this).
32 A. ABSTRACT HOMOTOPY THEORY
It is obvious that the fibrant objects of C loc are fibrant in C . Let us nowconsider the compositionHo( D ) fibrant replacement −−−−−−−−−−−−→ Ho( D f ) G −→ Ho( C loc ,f ) . Postcomposing with the equivalence Ho( C loc ,f ) ֒ → Ho( C loc ) yields the standardconstruction of the right derived functor R G associated to the Quillen equivalence( A. . G : Ho( D f ) → Ho( C loc ,f ) is essentially surjective. By Ken Brown’s Lemmaapplied to the Quillen adjunction ( A. . C loc are the same as the weak equivalences in C , so we can findfor any fibrant object X of C loc a zig-zag of C -weak equivalences to some GY with G ∈ D f , i.e. X is contained in the essential image of R G : Ho( D ) → Ho( C ).It remains to prove the converse, i.e. that any C -fibrant object X in the essentialimage of R G is also fibrant in C loc . For this we observe that, since Ho( C f ) → Ho( C )and Ho( D f ) → Ho( D ) are equivalences, X can be connected by a zig-zag of weakequivalences inside C f to GY for some Y ∈ D f . The latter is fibrant in C loc as( A. .
3) is a Quillen adjunction. Since C loc ,f ⊂ C f is closed under weak equivalencesas remarked above, also X is therefore fibrant in C loc , finishing the proof. (cid:3) In the main text we are usually concerend with model categories C in whichfiltered colimits are homotopical, i.e. for any filtered category I the colimit functor C I → C sends levelwise weak equivalences to weak equivalences. The followinglemma in particular tells us that this property is preserved under the localizationprocess of the above theorem: Lemma
A.2.6 . Let C be a model category in which filtered colimits are homo-topical, and let C loc be a Bousfield localization that is moreover cofibrantly generatedas a model category. Then filtered colimits in C loc are also homotopical. Proof.
Let I be a filtered category. As C loc is cofibrantly generated, theprojective model structure on ( C loc ) I exists, and with respect to this the colimitfunctor is left Quillen.If now f : X → Y is any weak equivalence in ( C loc ) I , then we can factor it intoan acyclic cofibration i followed by a fibration p , which is again acyclic by 2-out-of-3. As the acyclic fibrations in ( C loc ) I are defined levelwise, and since the acyclicfibrations of C loc and C agree, p is then in particular levelwise a weak equivalenceof C . The assumption therefore guarantees that colim I p is a weak equivalence (in C and hence also in C loc ). On the other hand, colim I being left Quillen impliesthat colim I i is an acyclic cofibration and hence in particular a weak equivalence in C loc . The claim then follows by another application of 2-out-of-3. (cid:3) Theorem A.2.5 has the disadvantage that the weak equivalences of C loc can behard to grasp as soon as the left adjoint F in ( A. .
2) is not homotopical. On theother hand, we usually have good control over the fibrant objects even if G is nothomotopical, so it will be useful to know how much this tells us about the modelstructure. Proposition
A.2.7 . Let C , C be model structures on the same underlyingcategory that have the same cofibrations as well as the same fibrant objects. Thenthe two model structures actually agree. .2. SOME HOMOTOPICAL ALGEBRA 233 Proof.
A proof can be found as [
Rie14 , Theorem 15.3.1], where this result isattributed to Joyal. (cid:3)
There is also a ‘relative’ version of this statement that needs some additionalassumptions:
Proposition
A.2.8 . Let C , D be simplicial model categories such that D is leftproper, and let (A.2.4) F : C ⇄ D : G be a simplicial adjunction such that F preserves cofibrations and G preserves fibrantobjects. Then ( A. . is already a Quillen adjunction. Proof.
This is [
Lur09 , Corollary A.3.7.2]. (cid:3)
A.2.2. Enlarging the class of cofibrations.
At several points, we want toenlarge the cofibrations of a given model structure, and in this section we devisetwo criteria for this. Both of them rely on the following characterization of certainmodel structures, which is due to Lurie building on previous work of Jeff Smith.Here the notion of a perfect class of morphisms occurs; while we can (and will)completely blackbox its definition here, the interested reader can find it as [
Lur09 ,Definition A.2.6.10].
Theorem
A.2.9 . Let C be a locally presentable category, let W be a class ofmorphisms in C , and let I be a set of morphisms in C . Then the following areequivalent: (1) There exists a (necessarily unique) left proper combinatorial model struc-ture on C with weak equivalences W and generating cofibrations I . More-over, filtered colimits are homotopical with respect to this model structure. (2) All of the following conditions are satisfied: (a)
The class W is perfect. (b) W is closed under pushouts along pushouts of morphisms of I . (c) If a morphism f in C has the right lifting property against I , then f ∈ W . Proof.
The implication (1) ⇒ (2) is [ Lur09 , Remark A.2.6.14], whereas(2) ⇒ (1) is [ Lur09 , Proposition A.2.6.13]. (cid:3)
Corollary
A.2.10 . Let C be a left proper combinatorial model category inwhich filtered colimits are homotopical. Let I be a set of generating cofibrationsand let I be any other set of morphisms in C such that weak equivalences in C areclosed under pushouts along pushouts of maps in I .Then there exists a (necessarily unique) combinatorial model structure on C with generating cofibrations I := I ∪ I and weak equivalences the weak equivalencesof C . This model structure is left proper and filtered colimits in it are combinatorial.It is right proper if the original one was. Proof.
Let us prove that the sets W and I satisfy the assumptions for theimplication ‘ ⇐ ’ of the previous theorem.We first observe that W is perfect by the other direction of the theorem, veri-fying Condition (a). Moreover, weak equivalences are closed under pushouts alongpushouts of maps in I (by left properness of the original model structure) andclosed under pushouts along pushouts of maps in I (by assumption), which proves
34 A. ABSTRACT HOMOTOPY THEORY
Condition (b). Finally, as I ⊃ I , any map with the right lifting property against I also has the right lifting property against I and hence is a weak equivalence,verifying Condition (c). This proves that the model structure exists, is left proper,and that filtered colimits in it are homotopical.It only remains to show that the new model structure is right proper providedthat the original one was. It is a general (and somewhat surprising) fact, thatbeing right proper only depends on the class of weak equivalences, see [ Rez02 ,Proposition 2.5]. However, in the present situation there is an easier argument:both model structures have the same weak equivalences and any cofibration in theold model structure is also a cofibration in the new one, so that any new fibration isalready a fibration in the old model structure. The claim thus follows immediatelyfrom the definition of right properness. (cid:3)
As a special case of this we can now ‘mix’ suitably nice model structures underonly a mild compatibility condition:
Corollary
A.2.11 . Let C , C be combinatorial model structures on the samecategory, such that every cofibration of C is also a cofibration in C . Assumemoreover that the weak equivalences of C are closed under filtered colimits andunder pushouts along cofibrations of C (!).Then there exists a (necessarily unique) mixed model structure C mix on thesame category whose weak equivalences are the weak equivalences of C and whosecofibrations are the cofibrations of C . This model structure is left proper, combi-natorial, and filtered colimits in it are homotopical. If C is right proper, then sois C mix . Proof. As C -cofibrations are in particular C -cofibrations, the assumptionsguarantee that C is left proper; moreover, we have explicitly assumed that filteredcolimits in it are homotopical.Now let I be any set of generating cofibrations of C . As the weak equivalencesof C are stable under pushouts along all cofibrations of C , they are in particularstable under pushouts along pushouts of I . We can therefore apply the previouscorollary to get a left proper combinatorial model structure with the same weakequivalences as C and with generating cofibrations I ∪ I , where I is our favouriteset of generating cofibrations of C . Moreover, the corollary asserts that this modelstructure is right proper if C was.To finish the proof it now suffices to identify this with the mixed model struc-ture, i.e. we have to prove that the cofibrations of this are precisely the cofibrationsof C . But indeed, I consists of cofibrations of C by definition, and so does I by the assumption that cofibrations in C are also cofibrations in C . Thus, any( I ∪ I )-cofibration is a cofibration in C . On the other hand, any cofibration in C is an I - and hence also a ( I ∪ I )-cofibration, finishing the proof. (cid:3) ibliography [AS69] Michael F. Atiyah and G. B. Segal. Equivariant K -Theory and Completion. J. Differ.Geom. , 3:1–18, 1969.[Ati61] Michael F. Atiyah. Characters and Cohomology of Finite Groups.
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