On the global homotopy theory of symmetric monoidal categories
aa r X i v : . [ m a t h . A T ] S e p ON THE GLOBAL HOMOTOPY THEORY OFSYMMETRIC MONOIDAL CATEGORIES
TOBIAS LENZ
Abstract.
Parsummable categories were defined by Schwede as the inputfor his global algebraic K -theory construction. We prove that their wholehomotopy theory with respect to the so-called global equivalences can alreadybe modelled by the more mundane symmetric monoidal categories.In another direction, we show that the resulting homotopy theory is alsoequivalent to the homotopy theory of a certain simplicial analogue of par-summable categories, that we call parsummable simplicial sets . These form abridge to several concepts of ‘globally coherently commutative monoids’ likeglobal Γ-spaces and ultracommutative monoids that will be explored in forth-coming work. Introduction
The algebraic K -theory of rings encodes information about a wide range of phe-nomena in number theory, geometry, and other areas of pure mathematics. Whilehistorically the roots of the subject lie in algebra, Quillen’s construction [Qui71] ofthe K -groups of a ring R is decidedly homotopy theoretic in nature: he first assignsto R an infinite loop space (or, in modern interpretation, a connective spectrum) K ( R ), and the K -groups are then only obtained in a second step as the homotopygroups of it.Quillen’s second construction [Qui73] made it clear that algebraic K -theory doesnot really depend on the ring R itself, but only on its module category. Building onthis observation, algebraic K -theory was soon extended to more general categoricalinputs. In particular, Shimada and Shimakawa [SS79] constructed the algebraic K -theory of small symmetric monoidal categories. More precisely, they show howsymmetric monoidal categories yield special Γ -spaces in the sense of Segal, whichwe can think of as ‘commutative monoids up to coherent systems of homotopies.’Segal’s delooping machinery [Seg74] then associates to each (special) Γ-space aconnective spectrum, and together this yields the K -theory K ( C ) of a symmetricmonoidal category C . If R is a ring, then applying K to a skeleton of the sym-metric monoidal category of finitely generated projective R -modules and R -linearisomorphisms under direct sum recovers the usual K -theory of R . K -theory as group completion. A particularly striking structural insight onthe K -theory of symmetric monoidal categories is Thomason’s result [Tho95, The-orem 5.1 and Lemma 1.9.2] that K exhibits the homotopy category of connectivespectra as localization of the category of small symmetric monoidal categories. Mathematics Subject Classification.
Primary 55P91, Secondary 19D23.
Key words and phrases.
Symmetric monoidal categories, parsummable categories, equivari-ant algebraic K -theory, global homotopy theory. This result was later refined by Mandell [Man10, Theorem 1.4], who showed thatalready the intermediate passage to the homotopy category of special Γ-spaces is alocalization, i.e. symmetric monoidal categories model all ‘coherently commutativemonoids in spaces.’ Thomason’s original result then follows from this via Segal’scomparison between the homotopy theories of (special) Γ-spaces and connectivespectra [Seg74, Proposition 3.4].More precisely, Segal shows (in modern language) that the passage from spe-cial Γ-spaces to connective spectra is a Bousfield localization and that it identifiesthe homotopy category of connective spectra with the one of the very special (or grouplike ) Γ-spaces. Together with Mandell’s result we can view this as a preciseformulation of the slogan that K -theory is ‘higher group completion,’ just like K can be defined as an ordinary group completion. Equivariant and global algebraic K -theory. The study of G -equivariant alge-braic K -theory for a fixed finite group G already began in the 80’s, but recent yearshave seen a renewed interest in it, for example through the work of Merling [Mer17]and her coauthors [MM19].The foundations of the subject were laid by Shimakawa [Shi89], who developedΓ - G -spaces as a G -equivariant generalization of Segal’s Γ-spaces, and used thismachinery to construct the equivariant K -theory K G ( C ) of a small symmetricmonoidal category C with a suitable G -action. Here K G ( C ) is a G -spectrum inthe sense of G -equivariant stable homotopy theory; we emphasize that this theoryis richer than the na¨ıve homotopy theory of G -objects in spectra, and similarlyfor Γ- G -spaces. In particular, Shimakawa’s result is not simply a consequence offunctoriality of the usual non-equivariant K -theory constructions—for example, wecan extract from K G ( C ) not only a single N -graded K -group, but in fact one foreach subgroup H ⊂ G , and these graded abelian groups are connected by additionalstructure maps providing them with the structure of a so-called G -Mackey functor .In this sense it turns out that the G -equivariant algebraic K -theory of a sym-metric monoidal category C with trivial G -action already contains interesting ad-ditional information. If we fix C and vary G , this yields a family of equivariant K -theory spectra associated to C , which are related by suitable change-of-groupmaps. A rigorous framework meant to capture the notion of such families is globalstable homotopy theory in the sense of [Sch18], and it is therefore natural to askwhether we can collect all this equivariant information in a single global spectrum .A candidate for this has been recently proposed by Schwede [Sch19b], who intro-duced global algebraic K -theory . His approach differs from the other constructionsdiscussed above in that it is not based on symmetric monoidal categories, but onso-called parsummable categories . However, there is a specific way to assign a par-summable category to a small symmetric monoidal category, which can then beused to define its global algebraic K -theory. New results.
The present article is a step towards refinements of Thomason’s andMandell’s results to equivariant and global algebraic K -theory.Firstly, we bring Schwede’s construction on an equal footing with the otherapproaches considered above by comparing their inputs. More precisely, there is anotion of global weak equivalences of parsummable categories, and global algebraic K -theory is invariant under them [Sch19b, Theorem 4.16]. On the other hand,Schwede introduced a global model structure on the category of small categories LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 3 (modelling unstable global homotopy theory) in [Sch19a], and we call a strongsymmetric monoidal functor a global weak equivalence if its underlying functoris a weak equivalence in this model structure. Our first main result can then beparaphrased as follows:
Theorem A (see Theorem 6.7) . The passage from small symmetric monoidal cat-egories to parsummable categories defines an equivalence of homotopy theories withrespect to the global weak equivalences (i.e. it induces an equivalence on the corre-sponding ∞ -categorical localizations). In particular, symmetric monoidal categories are just as good from the perspec-tive of global algebraic K -theory as general parsummable categories. This alsofollows the general pattern that on the pointset level global objects can often bemodelled by ordinary non-equivariant objects, and that it is only through the notionof weak equivalence that their equivariant behavior emerges.As our second contribution, we introduce parsummable simplicial sets as a sim-plicial analogue of parsummable categories. There is again a suitable notion ofglobal weak equivalences, and with respect to these we prove: Theorem B (see Theorem 5.8) . The nerve defines an equivalence of homotopytheories between the categories of parsummable categories and of parsummable sim-plicial sets.
We are particularly interested in parsummable simplicial sets because they forma bridge to several concepts of ‘globally coherently commutative monoids’ that westudy in the forthcoming [Len]. In particular, we will show that their homotopytheory is equivalent to a suitable global version of Γ-spaces and to an analogueof Schwede’s ultracommutative monoids [Sch18] for finite groups. These furthercomparisons use techniques from homotopical algebra, and in particular they usethat the global weak equivalences of parsummable simplicial sets are part of a modelstructure. It is not clear, whether such model structures also exist on the othercategories discussed in this article.Together with these comparisons, Theorems A and B above will then showthat symmetric monoidal categories model all ‘globally coherently commutativemonoids,’ yielding a global refinement of Mandell’s theorem and an ‘additive’version of Schwede’s result that categories model all unstable global homotopytypes [Sch19a, Theorem 3.3]. Together with a global version of Segal’s deloop-ing theory, that we also develop in [Len], this will in particular allow us to refineThomason’s original result to a global comparison. G -global homotopy theory. We will actually prove Theorems A and B in greatergenerality in this article: namely, we allow an additional discrete, but possiblyinfinite group G to act everywhere, and we consider the resulting categories of G -objects with respect to so-called G -global weak equivalences , which for G = 1recovers the previous definitions.However, for general G , the G -global weak equivalences are typically finer thanthe underlying global weak equivalences, and they are in particular fine enoughto recover the usual G -equivariant information when G is finite. More precisely,Shimakawa’s equivariant K -theory is invariant under G -global weak equivalences,so that the G -global versions of Theorems A and B provide structural information TOBIAS LENZ about the equivariant algebraic K -theory of G -symmetric monoidal categories. To-gether with the results of [Len] this will then also yield G -equivariant versions ofThomason’s and Mandell’s results. Outline.
In Section 1 we review the basic theory of tame E M -categories andparsummable categories, and we define the G -global homotopy theory of E M - G -categories and G -parsummable categories. Their simplicial counterparts are thenintroduced and studied in Section 2.Section 3 is devoted to the construction of the E M - G -category C X associatedto an E M - G -simplicial set X . We prove in Section 4 that the resulting functoris homotopy inverse to the nerve, and that the E M - G -categories arising this waysatisfy a certain technical condition that we call weak saturatedness . Section 5 isthen devoted to lifting the results of the previous two sections to G -parsummablecategories and G -parsummable simplicial sets, in particular proving Theorem B.Finally, we prove Theorem A in Section 6 by using the weak saturatedness es-tablished in Section 4 to reduce it to the categorical comparison result of [Len20]. Acknowledgements.
This article was written as part of my ongoing PhD thesisat the University of Bonn, and I would like to thank my advisor Stefan Schwedefor suggesting equivariant and global algebraic K -theory as a thesis topic, as wellas for helpful remarks on a previous version of this article. I am moreover indebtedto Markus Hausmann, who first suggested to me that there should be a notionof G -global homotopy theory. This proved to be a fruitful way of thinking aboutmany phenomena surrounding global K -theory, which will in particular permeatethe forthcoming [Len].I am grateful to the Max Planck Institute for Mathematics in Bonn for theirhospitality and 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).1. A Reminder on Parsummable Categories M -sets and tameness. We begin by recalling the monoid M as well as somebasic results about the combinatorics of M -actions from [SS20]. Definition 1.1. If A, B are sets, then we write Inj(
A, B ) for the set of injectivemaps A → B . We write ω for the countably infinite set { , , . . . } , and we write M for Inj( ω, ω ) considered as a monoid under composition. Warning . In [Sch19b, SS20], the notation M is used for the above monoid,while [Sch19b] uses M for what we call E M below. The reason for this changein notation is consistency with the forthcoming [Len], where actions of the abovemonoid are studied in their own right. In particular we prove in [Len] that simplicialsets with an action of Inj( ω, ω ) (when equipped with a slightly subtle notion of weakequivalence) already model global homotopy theory, so we think it deserves to benotationally distinguished from a generic monoid.The notation M for the above monoid has also been used in the literature before,for example in [Sch08]. LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 5
Next, we come to the notions of support and tameness for M -sets, whose cat-egorical and simplicial counterparts will later be central for defining parsummablecategories and parsummable simplicial sets, respectively. Definition 1.3.
Let X be an M -set, let x ∈ X , and let A ⊂ ω be finite. We saythat x is supported on A if u.x = x for all u ∈ M fixing A pointwise. We say that x is finitely supported if it is supported on some finite set. The M -set X is called tame if all its elements are finitely supported. Definition 1.4.
Let X be an M -set and let x ∈ X be finitely supported. Then the support supp( x ) of x is the intersection of all finite sets on which it is supported. Lemma 1.5.
In the above situation, x is supported on supp( x ) . Put differently, a finitely supported x is supported on a unique minimal set. Proof.
This is immediate from [SS20, Proposition 2.3], also see the discussion afterProposition 2.4 of op.cit. (cid:3)
Example . If A is any finite set, then we can consider Inj( A, ω ) with M -actiongiven by M ×
Inj(
A, ω ) , ( u, i ) u ◦ i . This is tame: an injection i : A → ω isobviously supported on i ( A ). In fact, supp( i ) = i ( A ): namely, if B i ( A ), thenwe can pick an a ∈ i ( A ) r B and an injection u fixing B pointwise with a / ∈ im u .Then u.i = u ◦ i does not hit a , so u.i = i , i.e. i is not supported on B . Non-example . If A is countably infinite, then Inj( A, ω ) is not tame, in particular M with its left regular action is not tame. In fact, no element is finitely supported:if i : A → ω is any injection, and B ⊂ ω is any finite set, then B i ( A ) as the righthand side is infinite. The same argument as in the previous example then showsthat i is not supported on B .We close this discussion by collecting some basic facts about the support for easyreference. Lemma 1.8. (1) If f : X → Y is an M -equivariant map of M -sets and x ∈ X is finitely supported, then also f ( x ) is finitely supported. Moreover, supp( f ( x )) ⊂ supp( x ) . (2) If X is an M -set and x ∈ X is finitely supported, then u.x is finitelysupported for all u ∈ M . Moreover, supp( u.x ) = u (supp( x )) .Proof. The first statement is immediate from the definition and it also appearswithout proof in [SS20, discussion after Proposition 2.5]. The second statementis [SS20, Proposition 2.5-(ii)]. (cid:3)
Lemma 1.9.
Let X be an M -set, let x ∈ X be supported on the finite set A ⊂ ω ,and let u, v ∈ M with u ( a ) = v ( a ) for all a ∈ A . Then u.x = v.x .Proof. This is [SS20, Proposition 2.5-(i)]. (cid:3) E M -categories and parsummable categories. We recall the so-called‘chaotic categories’:
Construction . Let X be a set. We write EX for the (small) category withset of objects Ob( EX ) = X and precisely one morphism x → y for any x, y ∈ X ,which we denote by ( y, x ). The composition is then uniquely determined by this;explicitly, ( z, y )( y, x ) = ( z, x ) for all x, y, z ∈ X . TOBIAS LENZ
For any map of sets f : X → Y there is a unique functor Ef : EX → EY thatis given on objects by f , and this way E becomes a functor Set → Cat . It is easyto check that E is right adjoint to Ob with counit Ob EX → X the identity.In particular, E preserves products and terminal objects, so it sends ordinarymonoids to monoids in the 1-category Cat (i.e. strict monoidal categories ). Centralto Schwede’s construction of global algebraic K -theory is the categorical monoid E M obtained this way. Definition 1.11. An E M -category is a category C together with a strict actionof E M . A map of E M -categories is a functor f : C → D strictly commuting withthe action, i.e. such that the diagram E M ×
C CE
M ×
D D act E M× f f act commutes. We write E M -Cat for the category of small E M -categories.If C is an E M -category, then we have in particular an action of the discretemonoid M on C , which then restricts to an M -action on the (large) set Ob C . Inaddition, we are given for each u, v ∈ M a natural isomorphism [ u, v ] : ( v. –) ⇒ ( u. –)given on x ∈ C by ( u, v ) . id x : v.x → u.x . Specializing to v = 1 this yields inparticular for each x ∈ C an isomorphism u x ◦ : x → u.x . From functoriality andassociativity of the action one easily concludes that(1.1) ( uv ) x ◦ = u v.x ◦ v x ◦ for all u, v ∈ M , x ∈ X . The following useful lemma shows that E M -actions oncategories can conversely be described by the above data, considerably simplifyingtheir construction: Lemma 1.12.
Let C be a category, and assume we are given an M -action on Ob C together with for each u ∈ M , x ∈ C an isomorphism u x ◦ : x → u.x such that thesedata satisfy the relation (1 . .Then there exists a unique E M -action on C extending the M -action on Ob C and such that u x ◦ = [ u, x for all x ∈ C and u ∈ M .Proof. See [Sch19b, Proposition 2.6]. (cid:3)
There is also a relative version of the lemma, that allows us to check E M -equivariance of functors in terms of the above data: Lemma 1.13.
Let
C, D be E M -categories and let f : C → D be a functor of theirunderlying categories. Then f is E M -equivariant if and only if Ob f : Ob C → Ob D is M -equivariant and f ( u x ◦ ) = u f ( x ) ◦ : f ( x ) → u. ( f ( x )) = f ( u.x ) for all x ∈ C , u ∈ M .Proof. This is [Len20, Corollary 1.3]. (cid:3)
We can now define the categorical counterpart of the support:
Definition 1.14.
Let C ∈ E M -Cat , let x ∈ C , and let A ⊂ ω be a finite set.Then we say that x is supported on A if x is supported on A as an element of the LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 7 M -set Ob( C ). We write C [ A ] ⊂ C for the full subcategory spanned by the objectssupported on A .We morever say that x is finitely supported if it is supported on some finite set,and we write C τ for the full subcategory of those.Finally, we call C tame if all its objects are finitely supported (i.e. if Ob( C )is tame), and we denote the full subcategory of E M -Cat spanned by the tame E M -categories by E M -Cat τ .Lemma 1.8-(1) shows that any E M -equivariant functor f : C → D restricts to C [ A ] → D [ A ] for all finite A ⊂ ω , and hence in particular to C τ → D τ . Definition 1.15.
Let C ∈ E M -Cat , and let x ∈ C be finitely supported. Thenthe support supp( x ) of x is the intersection of all (finite) sets A ⊂ ω on which x issupported, i.e. its support as an element of the M -set Ob( C ).Lemma 1.5 shows that x is indeed supported on supp( x ), also see [Sch19b, Propo-sition 2.13-(i)]. Example . If A is any finite set, then the M -action on Inj( A, ω ) from Ex-ample 1.6 induces an E M -action on E Inj(
A, ω ) recovering the original action onobjects. In particular, this E M -category is tame and the support of an object i : A → ω is precisely i ( A ).At first sight, the definition of support might seem a bit to weak (or ‘un-categorical’) as for u ∈ M fixing supp( x ) pointwise we only explicitly require that x = u.x and not that the canonical comparison isomorphism u x ◦ : x → u.x be theidentity. However, it turns out that this seemingly stronger condition is in factautomatically satisfied: Lemma 1.17.
Let C ∈ E M -Cat , let x ∈ C be finitely supported, and let u, v ∈ M agree on supp( x ) . Then u.x = v.x . If moreover u ′ , v ′ ∈ M agree on supp( x ) , then [ u ′ , u ] x = [ v ′ , v ] x : u.x = v.x → u ′ .x = v ′ .x .In particular, u x ◦ = id x if u restricts to the identity on supp( x ) .Proof. This follows from [Sch19b, Proposition 2.13-(ii)]. (cid:3)
Warning . The above lemma should not be misunderstood as saying that u.x = x for u ∈ M , x ∈ C if and only if u x ◦ is the identity. In particular, if H ⊂ M is a subgroup, then for x ∈ C H the structure isomorphisms h ◦ : x → h.x areusually non-trivial, and instead they define a potentially interesting H -action on x . In fact, for so-called saturated E M -categories C , that we will recall below,and ‘nicely’ embedded subgroups H , all H -objects in C arise this way, cf. [Sch19b,Construction 7.4].We are now ready to introduce the box product of tame E M -categories [Sch19b,Definition 2.31]: Definition 1.19.
Let
C, D ∈ E M -Cat τ . Their box product C ⊠ D is the fullsubcategory of C × D spanned by the pairs ( c, d ) of disjointly supported objects,i.e. pairs such that supp( c ) ∩ supp( d ) = ∅ .The box product is tame again [Sch19b, Corollary 2.33], and it becomes a sub-functor of the cartesian product. It is not hard to check that the usual unitality,associativity, and symmetry isomorphisms of the cartesian product restrict to corre-sponding isomorphisms for ⊠ , making it the tensor product of a preferred symmetric TOBIAS LENZ monoidal structure on E M -Cat τ with unit the terminal E M -category [Sch19b,Proposition 2.34]. Definition 1.20. A parsummable category is a commutative monoid for the boxproduct on E M -Cat τ . We write ParSumCat for the corresponding category ofcommutative monoids.Explicitly, this means that the data of a parsummable category consist of atame E M -category C equipped with an object 0 of empty support as well as asuitably functorial ‘addition’ defined for any pair of disjointly supported objects x, y ∈ C and for any pair of morphisms f : x → x ′ , g : y → y ′ such that x, y and x ′ , y ′ are disjointly supported. The addition has to be strictly associative,unital, and commutative whenever this makes sense. In general, two given objects x, y ∈ C might not be summable (i.e. supp( x ) ∩ supp( y ) = ∅ ), but [Sch19b, proofof Theorem 2.32] shows that we can always replace x, y by an isomorphic pair ofsummable objects.1.3. G -global homotopy theory of G -parsummable categories. For the restof this article, let us fix a discrete (not necessarily finite) group G . We will beinterested in the category E M - G -Cat of G -objects in E M -Cat , whose objectswe call small E M - G -categories. They can also be alternatively described as smallcategories with an E M -action and a G -action such that the two actions commute,or as small categories with an action of the categorical monoid E M × G .Any C ∈ E M - G -Cat has in particular an underlying E M -action, from which itinherits the notions of support and tameness. The full subcategory E M - G -Cat τ ⊂ E M - G -Cat of the tame E M - G -categories is then canonically identified with G -objects in E M -Cat τ .In particular, the box product of tame E M -categories together with its unital-ity, associativity, and symmetry isomorphisms automatically lifts to a symmetricmonoidal structure on E M - G -Cat τ . We write G -ParSumCat for the corre-sponding category of commutative monoids and call its objects G -parsummablecategories . The category G -ParSumCat is then again canonically identified withthe category of G -objects in ParSumCat .We now want to study E M - G -categories and G -parsummable categories froman equivariant point of view. The crucial observation for this is that any groupembeds into M in a particularly nice way; in order to describe these embeddingswe need the following two notions: Definition 1.21.
Let H be a finite group. A countable H -set X is called universal ,if every finite H -set embeds into X equivariantly. Definition 1.22.
A finite subgroup H ⊂ M is called universal if ω equipped withthe restriction of the tautological M -action to H is a universal H -set. Remark . The above notion of universality is analogous to [Sch20, Defini-tions 1.3 and 1.4] which studies global homotopy theory with respect to compactLie groups.
Lemma 1.24.
Any finite group H admits an injective homomorphism i : H → M with universal image. If j : H → M is another such homomorphism, then thereexists a (non-canonical) invertible element ϕ ∈ M with j ( h ) = ϕi ( h ) ϕ − for all h ∈ H . LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 9
This lemma appeared in a preliminary version of [Sch19b]. It tells us in particularthat we can associate to an E M -category C for any abstract finite group H anunderlying H -category well-defined up to (a priori non-canonical) isomorphism,explaining the global equivariant behaviour of E M -categories. ‘Proof.’ This is completely analogous to [Sch20, proof of Proposition 1.5], and weleave the details to the interested reader. (cid:3)
In order to define the ‘ G -global homotopy theory’ of small E M - G -categories,we introduce some notation: if H ⊂ M is a subgroup and ϕ : H → G is a homo-morphism, then we write Γ ϕ := { ( h, ϕ ( h )) : h ∈ H } for the corresponding graphsubgroup of E M × G . If C ∈ E M - G -Cat , then we abbreviate C ϕ := C Γ ϕ , and if f : C → D is a morphism in E M - G -Cat , then we write f ϕ := f Γ ϕ : C ϕ → D ϕ . Definition 1.25.
A morphism f : C → D in E M - G -Cat is called a G -globalweak equivalence if f ϕ : C ϕ → D ϕ is a weak homotopy equivalence (i.e. a weakhomotopy equivalence on nerves) for each universal subgroup H ⊂ M and eachhomomorphism ϕ : H → G .A morphism of G -parsummable categories is called a G -global weak equivalenceif it so as a morphism in E M - G -Cat .For G = 1, Schwede [Sch19b, Definition 2.26] considered these under the name‘global equivalence.’ Here we use the term ‘ G -global weak equivalence’ instead inorder to emphasize that these are a refinement of the weak homotopy equivalencesinstead of the categorical equivalences , i.e. those ( E M × G )-equivariant functorsthat are equivalences of underlying categories.In fact, the G -global weak equivalences and the categorical equivalences are ingeneral incomparable, i.e. neither notion implies the other. However, there is atleast a particular class of interesting E M - G -categories for which the G -global weakequivalences are indeed finer than the categorical equivalences, which we will nowintroduce. Construction . Let C be a (small) E M - G -category, let H ⊂ M be universal,and let ϕ : H → G be any group homomorphism. Then H acts on EH via its leftregular action, and it acts on C via the diagonal of its actions via M and ϕ . Wewrite C ‘ h ’ ϕ := Fun ϕ ( EH, C ) for the fixed points of the induced conjugation action.In other words, Fun ϕ ( EH, C ) is the subcategory of those ϕ : EH → C such that ϕ ◦ ( h. –) = (cid:0) ( h, ϕ ( h )) . – (cid:1) ◦ ϕ together with those natural transformations τ : ϕ ⇒ Ψsuch that τ h = ( h, ϕ ( h )) .τ .Restricting along EH → ∗ produces a fully faithful functor C → Fun(
EH, C )that is equivariant with respect to the above action on the target and H acting on C as before. In particular, we get an induced functor C ϕ → Fun ϕ ( EH, C ) = C ‘ h ’ ϕ that is again fully faithful as a limit of fully faithful functors.Here we use the notation C ‘ h ’ ϕ to stress that the above are homotopy fixed points,but with respect to the categorical equivalences and not with respect to the weakhomotopy equivalences or G -global weak equivalences. In particular, if f is anycategorical equivalence, then f ‘ h ’ ϕ will be an equivalence of categories, but if f is a G -global weak equivalence, then f ‘ h ’ ϕ need not be a weak homotopy equivalence. Definition 1.27.
A small E M - G -category C is called saturated if for all universal H ⊂ M and all ϕ : H → G the above functor C ϕ ֒ → C ‘ h ’ ϕ is an equivalence of categories. It is called weakly saturated if C ϕ ֒ → C ‘ h ’ ϕ is a weak homotopyequivalence.We write E M - G -Cat τ,s ⊂ E M - G -Cat τ,ws ⊂ E M - G -Cat τ for the full sub-categories of saturated and weakly saturated tame E M - G -categories, respectively.If G = 1, the above definition of saturatedness agrees with [Sch19b, Defini-tion 7.3]. However, for the present article the weak notion will be more important. Lemma 1.28.
Let f : C → D be a categorical equivalence in E M - G -Cat τ,ws .Then f is a G -global weak equivalence.Proof. Let H ⊂ M be a universal subgroup, and let ϕ : H → G be any grouphomomorphism. Then we have a commutative diagram C ϕ D ϕ C ‘ h ’ ϕ D ‘ h ’ ϕf ϕ f ‘ h ’ ϕ with the vertical maps as above. The bottom map is an equivalence of categoriesbecause f is, hence in particular a weak homotopy equivalence. Moreover, thevertical maps are weak homotopy equivalences by assumption, so that the tophorizontal arrow is also a weak homotopy equivalence by 2-out-of-3 as desired. (cid:3) We also recall our saturation construction that appeared for G = 1 as [Sch19b,Construction 7.18]: Construction . Let C be a small E M -category. Then Fun( E M , C ) carriestwo commuting E M -actions: one via the given action on C and one via the right E M -action on itself via precomposition. We equip Fun( E M , C ) with the diagonalof these two actions.Now assume that C is tame. We write C sat := Fun( E M , C ) τ . If f : C → D isan E M -equivariant functor, then we write f sat := Fun( E M , f ) τ ; we omit the easyverification that f sat is well-defined and E M -equivariant, and that this way (–) sat becomes an endofunctor of E M -Cat τ .Finally, we consider the E M -equivariant functor C → Fun( E M , C ) induced by E M → ∗ , which restricts to an E M -equivariant functor s : C → C sat , see [Sch19b,Theorem 7.22-(iv)]. We omit the easy verification that s is natural.Pulling through the G -actions everywhere, we can upgrade (–) sat to an endofunc-tor of E M - G -Cat τ , and s automatically defines a natural transformation from theidentity to this lift. Theorem 1.30.
Let C be a tame E M - G -category. (1) C sat is saturated, so that (–) sat restricts to a functor E M - G -Cat τ → E M - G -Cat τ, s . (2) s : C → C sat is a categorical equivalence.In particular, the inclusion E M - G -Cat τ, s ֒ → E M - G -Cat τ is a homotopy equiv-alence with respect to the categorical equivalences on both sides , and (–) sat ishomotopy inverse to it. Here we call a homotopical functor F : C → D of categories with weak equiv-alences a homotopy equivalence if there exists a homotopical functor G : D → C LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 11 that is homotopy inverse to it, i.e. such that both
F G and GF are connected byzig-zags of natural levelwise weak equivalences to the respective identities. Proof.
This is similar to the usual global situation, where this argument appearedin slightly different form as [Sch19b, Theorem 7.22].We will show that Fun( E M , C ) is saturated, and that the inclusion C sat =Fun( E M , C ) τ ֒ → Fun( E M , C ) induces an equivalence of categories on ϕ -fixedpoints for all universal H ⊂ M and all ϕ : H → G . Applying the latter to H = 1 inparticular shows that C sat ֒ → Fun( E M , C ) is a categorical equivalence, and sinceso is C → Fun( E M , C ) (as E M ≃ ∗ ), also s : C → C sat is a categorical equiva-lence by 2-out-of-3, proving (2). On the other hand, it shows that for any ϕ thetop horizontal and right hand vertical map in the evident commutative diagram( C sat ) ϕ Fun( E M , C ) ϕ ( C sat ) ‘ h ’ ϕ Fun( E M , C ) ‘ h ’ ϕ are equivalences of categories. Moreover, we can also deduce that the lower hori-zontal map is an equivalence of categories (as a homotopy limit of equivalences),hence so is the left hand vertical map by 2-out-of-3, which will precisely prove (1).It remains to prove the two claims. For the first claim, we have to show thatfor all H, ϕ as above the canonical map Fun( E M , C ) ϕ → Fun(
EH,
Fun( E M , C )) ϕ is an equivalence. Under the identification Fun( EH,
Fun( E M , C )) ∼ = Fun( EH × E M , C ) ∼ = Fun( E ( H × M ) , C ) given by the adjunction isomorphism and the factthat E preserves products, the right hand side of the above map corresponds tothe fixed points with respect to the same H -action on C as before and the H -action on E ( H × M ) induced by the left regular H -action and the H -action on M via h.u = uh − . Under this identification, the canonical map is induced by E (pr) : E ( H × M ) → E M , and it will be enough to show that this is an H -equivariant equivalence of categories (i.e. an equivalence in the 2-category of H -categories, H -equivariant functors, and H -equivariant natural transformations).For this we observe that both H × M and M are free H -sets. Thus, there existsan H -equivariant map r : M → H × M . It is then easy to check that for varying u ∈ M the unique maps pr( r ( u )) → u in E M assemble into an H -equivariantisomorphism E (pr) E ( r ) = E (pr ◦ r ) ∼ = id E M , and similarly E ( r ) E (pr) ∼ = id E ( H ×M ) equivariantly. This completes the proof of the first claim.For the second claim we observe that ( C sat ) ϕ → Fun( E M , C ) ϕ is always fullyfaithful as a limit of fully faithful functors, so that it is enough to show that it is alsoessentially surjective. Moreover, [Sch19b, Proposition 7.20] shows that Φ : E M → C is supported on some finite set A if (and only if) all Φ( u ) are supported on A and Φ factors through the restriction E (res) : E M → E Inj(
A, ω ).As H is universal, ω contains a finite faithful H -subset S (for example, wecould take any free H -orbit). By faithfulness, Inj( S, ω ) is a free H -set, hence thereexists an H -equivariant map χ : Inj( S, ω ) → H . As above one then argues that E ( χ ◦ res) ∗ : Fun( EH, C ) ϕ → Fun( E M , C ) ϕ is an equivalence of categories. More-over, the above characterization shows that it lands in Fun( E M , C ) τ = C sat ; moreprecisely, E ( χ ◦ res) ∗ (Φ) = E (res) ∗ E ( χ ) ∗ (Φ) is supported on S ∪ S h ∈ H supp(Φ( h )).We conclude that ( C sat ) ϕ ֒ → Fun( E M , C ) ϕ is essentially surjective, completing theproof of the second claim and hence of the theorem. (cid:3) Corollary 1.31.
The inclusion E M - G -Cat τ,s ֒ → E M - G -Cat τ,ws is a homotopyequivalence with respect to the G -global weak equivalences on both sides.Proof. We claim that (–) sat restricts to the desired homotopy inverse, for which itsuffices to show that s : C → C sat is a G -global weak equivalence for all weaklysaturated C . This is in turn an immediate consequence of the previous theoremtogether with Lemma 1.28. (cid:3) Next, we turn to parsummable structures. Schwede shows in [Sch19b, Construc-tion 7.23] (as an application of [Sch19b, Proposition 7.20] mentioned above) that thecanonical isomorphism Fun( E M , C ) × Fun( E M , D ) → Fun( E M , C × D ) restrictsfor any tame E M -categories C, D to a morphism C sat ⊠ D sat → ( C ⊠ D ) sat andthat this together with the unique map ∗ → ∗ sat makes (–) sat into a lax symmetricmonoidal functor with respect to ⊠ . In particular, if C is a parsummable category,then C sat again admits a natural parsummable structure: explicitly, the additionin C sat is given pointwise, and the unit for the addition is the functor constantat zero. The E M -equivariant functor s : C → C sat is then in fact a morphismof parsummable categories by [Sch19b, Theorem 7.25]. We therefore immediatelyconclude from the above: Corollary 1.32.
The inclusion G -ParSumCat s ֒ → G -ParSumCat of the fullsubcategory of saturated parsummable categories is a homotopy equivalence withrespect to the categorical equivalences on both sides. A homotopy inverse is givenby the above saturation construction. (cid:3) Corollary 1.33.
The inclusion G -ParSumCat s ֒ → G -ParSumCat ws is a ho-motopy equivalence with respect to the G -global weak equivalences on both sides. (cid:3) We will prove in Theorem 5.9 below that also the inclusion G -ParSumCat ws ֒ → G -ParSumCat is a homotopy equivalence with respect to the G -global weak equiv-alences, or in other words, that G -ParSumCat s ֒ → G -ParSumCat is a homotopyequivalence not only with respect to the categorical equivalences, but also with re-spect to the G -global weak equivalences. We emphasize again that C → C sat is usually not a G -global weak equivalence, so that this is not a consequence ofCorollary 1.32.2. E M -Simplicial Sets and Parsummable Simplicial Sets Also the functor (–) : SSet → Set admits a right adjoint, that we again de-note by E . Explicitly, ( EX ) n = Q ni =0 X with the evident functoriality in the twovariables. By uniqueness of adjoints we can then conclude that the simplicial set EX is canonically isomorphic to the nerve of the category EX ; more precisely,there is a unique simplicial map that is the identity on vertices, and this map is anisomorphism.As before, we see that E : Set → SSet preserves products, and hence sendsmonoids to simplicial monoids; in particular, we get a monoid E M , which by theabove is identified with N( E M ) with monoid structure induced by the categoricalmonoid E M . Definition 2.1. An E M -simplicial set is a simplicial set X together with an actionof the simplicial monoid E M , i.e. a simplicial map E M× X → X that is associativeand unital. We write E M -SSet for the category of E M -simplicial sets and E M -equivariant simplicial maps. LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 13 If C is a small E M -category, then N( C ) inherits an E M -action via E M × N( C ) ∼ = N( E M ) × N( C ) ∼ = N( E M × C ) N(act) −−−−→ N( C )where the first isomorphism is the one discussed above and the second one comesfrom the fact that N preserves finite products. This way, the nerve obviously liftsto a functor E M -Cat → E M -SSet . Remark . Let C be a small E M -category. We want to make the above E M -action on N( C ) explicit.For this let us write ⋄ for the action functor E M × C → C . Then we cancalculate for all u, v ∈ M and f : x → y in C :( v, u ) ⋄ f = (cid:0) ( v, u ) ◦ id u (cid:1) ⋄ (id y ◦ f ) = (cid:0) ( v, u ) ⋄ id y (cid:1) ◦ (id u ⋄ f ) = [ v, u ] y ◦ u.f ;as u.f = u y ◦ f ( u x ◦ ) − we conclude from this that the diagram x yu.x v.y u x ◦ f v y ◦ ( v,u ) ⋄ f commutes. Since the vertical maps are isomorphisms, this in fact completely de-termines ( v, u ) ⋄ f . We can therefore immediately conclude that the action of( u , . . . , u k ) ∈ ( E M ) k on a k -simplex x α −→ x → · · · → x k is uniquely character-ized by demanding that inserting it as the lower row in x x · · · x k u .x u .x · · · u k .x kα ( u ) x ◦ ( u ) x ◦ ( u k ) xk ◦ makes all the squares commute. Example . Let A be a (finite) set. Analogously to Example 1.16, the canoncial M -action on Inj( A, ω ) makes E Inj(
A, ω ) into an E M -simplicial set. It is thencanonically isomorphic to the nerve of the category of the same name consideredin the aforementioned example.2.1. Supports and tameness.
Next, we want to introduce analogues of the no-tions of support and tameness for E M -simplicial sets. Construction . Let 0 ≤ k ≤ n . We write i k : M → M n +1 for the homomorphismsending u ∈ M to (1 , . . . , , u, . . . ,
1) where u is in the ( k + 1)-th spot.If X is an E M -simplicial set, we therefore get ( n + 1) commuting M -actions on X n by pulling back the action of M n +1 = ( E M ) n along the injections i , . . . , i n . Definition 2.5.
Let X be an E M -simplicial set, let 0 ≤ k ≤ n , and let x ∈ X n .Then we say that x is k -supported on the finite set A ⊂ ω if it is supported on A as an element of the M -set i ∗ k X n . We say that x is k -finitely supported if it is k -supported on some finite set, in which case we write supp k ( x ) for its support in i ∗ k X n .We say that x is supported on A if it is k -supported on A for all 0 ≤ k ≤ n ,and we call it finitely supported if it is supported on some finite set, i.e. if it is k -finitely supported for all 0 ≤ k ≤ n . In this case its support supp( x ) is defined as S nk =0 supp k ( x ). The E M -simplicial set X is called tame if all its simplices are finitely supported.We write E M -SSet τ ⊂ E M -SSet for the full subcategory spanned by the tame E M -simplicial sets. Remark . Any E M -simplicial set X in particular forgets to an M -simplicialset, and it is natural to ask whether we can characterize the support of x ∈ X n interms of the resulting M -action (i.e. the diagonal action) on X n , analogously toLemma 1.17.We prove in [Len] that this is indeed the case: x is supported on the finite set A in the above sense if and only if it supported on A as an element of the M -set X n .However, the combinatorial argument for this is somewhat lengthy, and as we willnot need this result for the present article, we have decided to omit it. Example . Let C be a tame E M -category. We claim that an n -simplex α • :=( x α −→ x → · · · → x n ) of N( C ) is k -supported on the finite set A ⊂ ω if and onlyif A ⊃ supp( x k ), i.e. if x k is supported on A . In particular this shows that α • isfinitely supported and supp k ( α • ) = supp( x k ), supp( α • ) = S nk =0 supp( x k ).To prove the claim let us first assume that A contains supp( x k ); we will showthat i k ( u ) .α • = α • for all u fixing A pointwise. Indeed, by the description of( u , . . . , u n ) .α • from Remark 2.2 it suffices that each ( u k ) x k ◦ is the identity, whichis immediate from Lemma 1.17.Conversely, assume α • is k -supported on A , and let u be any injection fixing A pointwise, so that in particular i k ( u ) .α • = α • . Comparing the k -th vertices ofthese n -simplices then shows that u.x k = x k , and letting u vary we see that x k issupported on A as desired.In particular, we conclude from the above example that the nerve restricts to E M -Cat τ → E M -SSet τ . Example . Let A be a finite set. Then the above together with Example 1.16shows that the E M -simplicial set E Inj(
A, ω ) from Example 2.3 is tame and thatsupp k ( i , . . . , i n ) = i k ( A ). Warning . Example 2.7 shows that if X is isomorphic to the nerve of a tame E M -category, then the k -th support of an n -simplex x agrees with the support ofits k -th vertex. This is not true for general tame E M -simplicial sets. Even worse,the support of an n -simplex can be strictly larger than the union of the supportsof its vertices, for which we will give an example now:Let A be a non-empty finite set, and let X be obtained from E Inj(
A, ω ) × ∆ by collapsing both copies of E Inj(
A, ω ) to a single point. This is still tame sinceany n -simplex of X can be represented by some n -simplex of E Inj(
A, ω ) × ∆ , onwhose support it is then obviously supported. Moreover, the unique vertex of X has empty support for trivial reasons.However, the quotient does not identify any two edges { i } × ∆ , { j } × ∆ fordistinct injections i, j : A → ω . In particular, supp[ { i } × ∆ ] = i ( A ) = ∅ . Lemma 2.10.
Let f : X → Y be an E M -equivariant map of E M -simplicial sets,let ≤ k ≤ n , and let x ∈ X n be k -supported on some finite set A ⊂ ω . Then also f ( x ) is k -supported on A .In particular, if x is supported on A , then so is f ( x ) .Proof. The first statement is an instance of Lemma 1.8-(1), and the second onefollows immediately from this. (cid:3)
LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 15
Lemma 2.11.
Let X be an E M -simplicial set, and let x ∈ X n be k -finitely sup-ported for some ≤ k ≤ n . Then ( u , . . . , u n ) .x is k -finitely supported for all u , . . . , u n ∈ M , and supp k (( u , . . . , u n ) .x ) ⊂ u k (supp k ( x )) .Proof. The set map ( u , . . . , u n ) . – : X n → X n factors as(( u , . . . , u k − , , u k +1 , . . . , u n ) . –) ◦ ( i k ( u k ) . –) . As a self map of the M -set i ∗ k X n , the former is M -equivariant, so the claim followsfrom the two parts of Lemma 1.8. (cid:3) Lemma 2.12.
Let X be a tame E M -simplicial set, let x ∈ X n , and let ϕ : [ m ] → [ n ] be any map in ∆ . Then supp k ( ϕ ∗ x ) ⊂ supp ϕ ( k ) ( x ) for all ≤ k ≤ m .Proof. Let b ∈ ω r supp ϕ ( k ) ( x ) and let u be an injection fixing supp ϕ ( k ) ( x ) pointwisewith b / ∈ im( u ). Then i ϕ ( k ) ( u ) .x = x , hence( i ϕ ( k ) ( u ) ϕ (0) , . . . , i ϕ ( k ) ( u ) ϕ ( m ) ) .ϕ ∗ x = ϕ ∗ ( i ϕ ( k ) ( u )) .ϕ ∗ x = ϕ ∗ ( i ϕ ( k ) ( u ) .x ) = ϕ ∗ ( x ) . As ϕ ∗ x is k -finitely supported, we conclude from the previous lemma thatsupp k ( ϕ ∗ x ) = supp k (( i ϕ ( k ) ( u ) ϕ (0) , . . . , i ϕ ( k ) ( u ) ϕ ( m ) ) .ϕ ∗ x ) ⊂ i ϕ ( k ) ( u ) ϕ ( k ) (supp k ϕ ∗ x ) = u (supp k ϕ ∗ x ) , hence in particular b / ∈ supp k ( ϕ ∗ x ). The claim follows by letting b vary. (cid:3) Warning . The above argument presupposes that ϕ ∗ x is k -finitely supported,so it does not show that the finitely supported simplices of a general E M -simplicialset form a subcomplex. While this does indeed hold (see for example Remark 2.6above), the proof is harder and as we will only be interested in tame E M -simplicialsets below, we have decided to only consider the slightly weaker version above.We can now prove the following analogue of Lemma 1.17: Lemma 2.14.
Let X be a tame E M -simplicial set, let x ∈ X n be supported onthe finite set A ⊂ ω , and let ϕ : [ m ] → [ n ] be a map in ∆ . Then ( u , . . . , u n ) .ϕ ∗ x =( v , . . . , v n ) .ϕ ∗ x for all u , . . . , u n , v , . . . , v n ∈ M such that u i ( a ) = v i ( a ) for all i = 0 , . . . , n and a ∈ A .Proof. The previous lemma immediately implies that ϕ ∗ x is supported on A , so wemay assume without loss of generality that m = n and ϕ = id. Using Lemma 1.9together with Lemma 2.11, one then easily shows by descending induction that(1 , . . . , , u k , . . . , u n ) .x = (1 , . . . , , v k , . . . , v n ) .x for all 0 ≤ k ≤ n + 1, which for k = 0 is precisely what we wanted to prove. (cid:3) Proposition 2.15.
Let A ⊂ ω be finite, and let X be a tame E M -simplicialset. Then the simplices of X supported on A form a subcomplex X [ A ] ⊂ X , and (–) [ A ] : E M -SSet τ → SSet is a subfunctor of the forgetful functor.Moreover, it is corepresentable in the enriched sense by the E M -simplicial set E Inj(
A, ω ) from Example 2.3 via evaluation at the inclusion ι A : A ֒ → ω .Proof. We first observe that E Inj(
A, ω ) is tame and that ι A is supported on A byExample 2.8. We will show that the simplicial map ev : maps E M ( E Inj(
A, ω ) , X ) → X given by evaluation at ι A is injective with image precisely X [ A ] for each E M -simplicial set X . All the remaining claims will then easily follow from this. Let us show that the evaluation is injective. Indeed, by definition a k -simplex ofmaps E M ( E Inj(
A, ω ) , X ) is given by an E M -equivariant simplicial map E Inj(
A, ω ) × ∆ n → X and we have to show that any two such maps f, g whose restrictions to { ι A } × ∆ n agree, are already equal. But indeed, for any k -simplex (( u , . . . , u k ) , ϕ ), f (( u , . . . , u k ) , ϕ ) = f (cid:0) (ˆ u , . . . , ˆ u n ) . ( ι T , ϕ ) (cid:1) = (ˆ u , . . . , ˆ u n ) .f ( ι T , ϕ )= (ˆ u , . . . , ˆ u n ) .g ( ι T , ϕ ) = · · · = g (( u , . . . , u k ) , ϕ )where ˆ u i is any extension of u i to an injection ω → ω and we confuse the vertex ι T with its degeneracies. This proves injectivity.Lemma 2.10 shows that ev( f ) is supported on A for all f : E Inj(
A, ω ) × ∆ n → X .Conversely, let x be an n -simplex supported on A . Then Lemma 2.14 shows thatthe assignment f x : E Inj(
A, ω ) × ∆ n → X (( u k , . . . , u k ) , ϕ ) (ˆ u , . . . , ˆ u k ) .ϕ ∗ x is independent of the chosen extensions ˆ u i ∈ M . From this it easily follows that f x is simplicial, E M -equivariant, and that ev f x = x , proving surjectivity. (cid:3) In particular, we see that any map α : K → X [ A ] admits a unique E M -equivariantextension E Inj(
A, ω ) × K → X ; we will usually denote this extension by ˜ α .2.2. Parsummable simplicial sets.
We are now ready to introduce the box prod-uct of tame E M -simplicial sets: Construction . Let
X, Y be tame E M -simplicial sets, and let n ≥
0. Wedefine ( X ⊠ Y ) n ⊂ ( X × Y ) n to consist of precisely those pairs ( x, y ) such thatsupp k ( x ) ∩ supp k ( y ) = ∅ for all 0 ≤ k ≤ n . Proposition 2.17.
Let
X, Y be tame E M -simplicial sets. Then the above definesan E M -simplicial subset X ⊠ Y ⊂ X × Y , which we call the box product of X and Y . Both X ⊠ Y and X × Y are tame, and – ⊠ – is a subfunctor E M -SSet τ × E M -SSet τ → E M -SSet τ of the cartesian product.Proof. Lemma 2.12 shows that X ⊠ Y is a subcomplex, and it is closed under the(diagonal) E M -action by Lemma 2.11. Moreover, if f : X → X ′ and g : Y → Y ′ are E M -equivariant, then ( f × g )( X ⊠ Y ) ⊂ X ′ ⊠ Y ′ by Lemma 2.10.It only remains to show that X × Y (and hence X ⊠ Y ) is tame, for which itsuffices to observe that ( x, y ) is by definition k -supported on supp k ( x ) ∪ supp k ( y )(in fact, supp k ( x, y ) = supp k ( x ) ∪ supp k ( y )) for all x ∈ X n , y ∈ Y n , and 0 ≤ k ≤ n ,and hence in particular supported on supp( x ) ∪ supp( y ) (in fact, supp( x, y ) =supp( x ) ∪ supp( y )). (cid:3) Proposition 2.18.
The unitality, associativity, and symmetry isomorphisms ofthe cartesian product on E M -SSet τ restrict to corresponding isomorphisms for ⊠ .This makes E M -SSet τ into a symmetric monoidal category with tensor product ⊠ and unit the terminal E M -simplicial set.Proof. We will show that the associativity isomorphism ( X × Y ) × Z → X × ( Y × Z )restricts to an isomorphism ( X ⊠ Y ) ⊠ Z → X ⊠ ( Y ⊠ Z ) for all X, Y, Z ∈ E M -SSet τ . LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 17
Indeed, we have to show that if x ∈ X, y ∈ Y, z ∈ Z , then (( x, y ) , z ) ∈ ( X ⊠ Y ) ⊠ Z if and only if ( x, ( y, z )) ∈ X ⊠ ( Y ⊠ Z ). But the first condition is equivalent todemanding that supp k ( x ) ∩ supp k ( y ) = ∅ and supp k ( x, y ) ∩ supp k ( z ) = ∅ . We haveseen in the proof of the previous proposition that supp k ( x, y ) = supp k ( x ) ∪ supp k ( y ),so these two together are equivalent to demanding that supp k ( x ) , supp k ( y ) , supp k ( z )be pairwise disjoint. By a symmetric argument this is then in turn equivalent to( x, ( y, z )) ∈ X ⊠ ( Y ⊠ Z ) as desired.The arguments for the symmetry and unitality isomorphisms are similar, and weomit them. All the necessary coherence conditions of the resulting isomorphismsthen follow automatically from the corresponding results for the cartesian symmet-ric monoidal structure. (cid:3) Definition 2.19. A parsummable simplicial set is a commutative monoid for ⊠ in E M -SSet τ . We write ParSumSSet for the corresponding category.
Proposition 2.20.
The canonical isomorphism N( C ) × N( D ) → N( C × D ) restrictsto an isomorphism N( C ) ⊠ N( D ) → N( C ⊠ D ) . Together with the unique map ∗ → N( ∗ ) this makes N : E M -Cat τ → E M -SSet τ into a strong symmetric monoidalfunctor with respect to the box products on both sides.Proof. Let us prove the first statement, which amounts to saying that if x α −→ x → · · · x n and y β −→ y → · · · y n are n -simplices of N( C ) and N( D ), respectively, then(2.1) ( x , y ) ( α ,β ) −−−−−→ ( x , y ) → · · · → ( x n , y n )lies in the image of N( C ⊠ D ) → N( C × D ) if and only if ( α • , β • ) ∈ N( C ) ⊠ N( D ).But indeed, the latter condition is equivalent to supp k ( α • ) ∩ supp k ( β • ) = ∅ for all0 ≤ k ≤ n , which by Example 2.7 is further equivalent to supp( x k ) ∩ supp( y k ) = ∅ for all 0 ≤ k ≤ n . But this is by definition equivalent to ( x k , y k ) ∈ C ⊠ D for all0 ≤ k ≤ n , which is in turn equivalent to ( α k , β k ) : ( x k − , y k − ) → ( x k , y k ) being amorphism in C ⊠ D for 1 ≤ k ≤ n as C ⊠ D ⊂ C × D is a full subcategory. Finally,by definition of the nerve this is further equivalent to (2 .
1) lying in the image ofN( C ⊠ D ) → N( C × D ), which completes the proof of the first statement.It is clear that also ∗ → N( ∗ ) is an isomorphism. As all the structure iso-morphisms on both E M -Cat τ and E M -SSet τ are defined as restrictions of thestructure isomorphisms of the cartesian symmetric monoidal structures, all the nec-essary coherence conditions hold automatically, which completes the proof of theproposition. (cid:3) In particular, we see that the nerve lifts to
ParSumCat → ParSumSSet .Explicitly, this sends a parsummable category C to N( C ) with E M -action as above.The additive unit is given by the vertex 0 ∈ C , and if x α −→ x → · · · → x n and y β −→ y → · · · → y n are summable n -simplices, then supp( x k ) ∩ supp( y k ) for 0 ≤ k ≤ n , and α • + β • isthe n -simplex ( x + y ) α + β −−−−→ ( x + y ) → · · · → ( x n + y n ) . G -global homotopy theory of G -parsummable simplicial sets. We re-mind the reader that we fixed a discrete group G . As before, we can extend thebox product formally to the category E M - G -SSet τ of G -objects in E M -SSet τ ,i.e. tame E M -simplicial sets with a G -action through E M -equivariant morphisms,which we can further identify with simplicial sets with an action of the simplicialmonoid E M × G , so that the underlying E M -simplicial set is tame.The category G -ParSumSSet of commutative monoids in E M - G -SSet τ isthen canonically identified with the G -objects in ParSumSSet . Moreover, thenerve lifts to a strong symmetric monoidal functor E M - G -Cat τ → E M - G -SSet τ inducing N : G -ParSumCat → G -ParSumSSet . We now want to consider thesefrom a G -global perspective: Definition 2.21.
A morphism f : X → Y in E M - G -SSet is called a G -globalweak equivalence if f ϕ : X ϕ → Y ϕ is a weak homotopy equivalence for all universalsubgroups H ⊂ M and all homomorphisms ϕ : H → G .Here we again write (–) ϕ for the fixed points with respect to the graph subgroupΓ ϕ ⊂ M × G . Definition 2.22.
A morphism f : X → Y in G -ParSumSSet is called a G -globalweak equivalence if its underlying morphism of E M - G -simplicial sets is. Remark . As the nerve is a right adjoint, it commutes with taking ϕ -fixed points(up to canonical isomorphism). In particular, N : E M - G -Cat → E M - G -SSet preserves and reflects weak equvialences, and likewise for N : G -ParSumCat → G -ParSumSSet .3. The E M -Category Associated to an E M -Simplicial Set While N :
Cat → SSet is a homotopy equivalence, its left adjoint h (sending asimplicial set to its homotopy category ) is not homotopically meaningful. Instead,a possible homotopy inverse (going back to Quillen) of the nerve is the following:
Definition 3.1.
Let X be a simplicial set. Its category of simplices ∆ ↓ X is thesmall category with objects the simplicial maps f : ∆ n → X ( n ≥
0) and morphisms α : f → g those simplicial maps α satisfying f = g ◦ α .If S ⊂ [ m ], let us write ∆ S for the unique ( | S | − m with set ofvertices S . Construction . Let X be simplicial set. A general k -simplex α • of N(∆ ↓ X )then corresponds to a diagram∆ n ∆ n · · · ∆ n k .X α f α f α k f k There is a unique k -simplex σ α • of ∆ n k with ℓ -th vertex (0 ≤ ℓ ≤ k ) given by α k · · · α ℓ +1 (∆ { n ℓ } ) as ∆ n k is the nerve of a poset and since ∆ { n ℓ +1 } ≥ α ℓ +1 (∆ { n ℓ } )in ∆ n ℓ for all ℓ = 0 , . . . , k − ǫ : N(∆ ↓ X ) → X via ǫ ( α • ) := f k ( σ α • ).One can show that ǫ is indeed a simplicial map, and that it is natural withrespect to the functoriality of ∆ ↓ – via postcomposition. If X is the nerve of a LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 19 category, the above construction appears in [Ill72, VI.3], while the general versionseems to originate with Thomason [Tho95, Proposition 4.2].
Proposition 3.3.
For any simplicial set X the ‘last vertex map’ ǫ : N(∆ ↓ X ) → X is a weak homotopy equivalence.Proof. Thomason proves the topological analogue of this as [Tho95, Proposition 4.2];unfortunately, this does not immediately imply the above simplicial version becauseit is not clear a priori that N(∆ ↓ –) preserves weak homotopy equivalences.Instead, we will use that the last vertex map is an ∞ -categorical localization, seee.g. [Cis19, Proposition 7.3.15]. As any ∞ -categorical localization is in particulara weak homotopy equivalence, this immediately implies the proposition. (cid:3) One crucial step [Tho95, Proposition 4.5] in Thomason’s comparison betweensymmetric monoidal categories and connective spectra is a variant of the aboveconstruction yielding a functor from E ∞ spaces to lax symmetric monoidal cat-egories. Similarly, our proofs of Theorems A and B will rely on a parsummablerefinement C • of it. The rest of this section is devoted to constructing the under-lying E M -category of this together with an analogue of the ‘last vertex map.’ Construction . Let X be an E M -simplicial set. We define a small category C X as follows: an object of C X is a quadruple ( A, S, m • , f ) consisting of two finitesubsets A, S ⊂ ω , a family ( m a ) a ∈ A of non-negative integers m a ≥
0, and an E M -equivariant map f : E Inj(
S, ω ) × Q a ∈ A ∆ m a → X , where E M acts on E Inj(
S, ω )as in Example 2.3. A morphism (
A, S, m • , f ) → ( B, T, n • , g ) is an E M -equivariantmap α : E Inj(
S, ω ) × Q a ∈ A ∆ m a → E Inj(
T, ω ) × Q b ∈ B ∆ n b such that gα = f .Composition is inherited from the composition in E M -SSet ; in particular, theidentity of ( A, S, m • , f ) is given by the identity of E Inj(
S, ω ) × Q a ∈ A ∆ m a .We now define for each u ∈ M and each object ( A, S, m • , f ) of C X the ob-ject u. ( A, S, m • , f ) as the quadruple ( u ( A ) , u ( S ) , m u − ( • ) , f ◦ ( u ∗ × u ∗ )) where( m u − ( • ) ) b = m u − ( b ) for each b ∈ u ( A ), u ∗ : E Inj( u ( S ) , ω ) → E Inj(
S, ω ) is restric-tion along u : S → u ( S ), and u ∗ : Q b ∈ u ( A ) ∆ m u − b ) → Q a ∈ A ∆ m a is the uniquemap with pr a ◦ u ∗ = pr u ( a ) .Finally, we define u ( A,S,m • ,f ) ◦ : ( A, S, m • , f ) → u. ( A, S, m • , f ) as( u ∗ × u ∗ ) − : E Inj(
S, ω ) × Y a ∈ A ∆ m a → E Inj( u ( S ) , ω ) × Y b ∈ u ( A ) ∆ m u − b ) . Warning . Strictly speaking, the above description of C X is not entirely precisesince the hom-sets are not pairwise disjoint as one would usually require. Of course,this is not an actual issue because we can always make them disjoint artificially,which we will simply assume from now on. However, it means that in order toestablish an equality of morphisms in C X it is not enough to show the equality in E M -SSet , but instead we also have to check that their sources and targets agree. Lemma 3.6.
The above defines an E M -action on C X .Proof. It is clear that u ∗ × u ∗ is E M -equivariant, so that ( u ( A ) , u ( S ) , m u − ( • ) , f ◦ ( u ∗ × u ∗ )) is again an object of C X . Moreover, it is clearly an isomorphism, so that( u ∗ × u ∗ ) − is well-defined and again E M -equivariant; as it tautologically commuteswith the reference maps to X , we see that u ( A,S,m • ,f ) ◦ is indeed an isomorphism( A, S, m • , f ) → u. ( A, S, m • , f ). To finish the proof it suffices that the above defines an M -action on Ob( C X )and that(3.1) u v. ( A,S,m • ,f ) ◦ v ( A,S,m • ,f ) ◦ = ( uv ) ( A,S,m • ,f ) ◦ for all u, v ∈ M and ( A, S, m • , f ) ∈ C X .It is clear from the definition that 1 . ( A, S, m • , f ) = ( A, S, m • , f ). Moreover, oneeasily checks that the diagram E Inj(( uv )( S ) , ω ) × Q c ∈ ( uv )( A ) ∆ m ( uv ) − c ) E Inj(
S, ω ) × Q a ∈ A ∆ m a E Inj( u ( v ( S )) , ω ) × Q c ∈ u ( v ( A )) ∆ ( m v − • ) ) u − c ) E Inj( v ( S ) , ω ) × Q b ∈ v ( A ) ∆ m v − b ) ( uv ) ∗ × ( uv ) ∗ u ∗ × u ∗ v ∗ × v ∗ commutes, which immediately implies the associativity of the M -action. Moreover,it shows that the identity (3 .
1) holds as morphisms in E M -SSet ; as both sides aremorphisms ( A, S, m • , f ) → ( uv ) . ( A, S, m • , f ) = u. ( v. ( A, S, m • , f )), they then alsoagree as morphisms in C X , which completes the proof of the lemma. (cid:3) Lemma 3.7.
The E M -category C X is tame. Moreover, supp( A, S, m • , f ) = A ∪ S for any object ( A, S, m • , f ) ∈ C X .Proof. Let us first show that (
A, S, m • , f ) is supported on A ∪ S , which will in par-ticular imply tameness of C X . If u fixes A and S pointwise, then obviously u ( A ) = A, u ( S ) = S and m u − ( • ) = m • . Moreover, it is clear from the definition that both u ∗ : E Inj( u ( S ) , ω ) → E Inj(
S, ω ) and u ∗ : Q a ∈ A ∆ m a → Q b ∈ u ( A ) ∆ m u − b ) are therespective identities, so f ◦ ( u ∗ × u ∗ ) = f , and hence altogether u. ( A, S, m • , f ) =( A, S, m • , f ) as desired.Conversely, let ( A, S, m • , f ) be supported on some finite set B ; we have to showthat A ⊂ B and S ⊂ B . We will only prove the first statement (the argument forthe second one being analogous), for which we argue by contradiction: if A B ,then we choose any a ∈ A r B and an injection u fixing B pointwise such that a / ∈ im u . But then a / ∈ u ( A ), hence u ( A ) = A and u. ( A, S, m • , f ) = ( A, S, m • , f )contradicting the assumption that ( A, S, m • , f ) be supported on B . (cid:3) Construction . Let ϕ : X → Y be an E M -equivariant map. We define C ϕ : C X → C Y as follows: an object ( A, S, m • , f ) is sent to ( A, S, m • , ϕ ◦ f ) and a morphism( A, S, m • , f ) → ( B, T, n • , g ) given by α : E Inj(
S, ω ) × Q a ∈ A ∆ m a → E Inj(
T, ω ) × Q b ∈ B ∆ n b is sent to the morphism ( A, S, m • , ϕ ◦ f ) → ( B, T, n • , ϕ ◦ g ) given by thesame α . Lemma 3.9.
In the above situation, C ϕ is an E M -equivariant functor. Thisdefines a functor C • : E M -SSet → E M -Cat τ .Proof. It is clear that C ϕ is a well-defined functor and that it commutes with the M -action on objects. To show that it is E M -equivariant, it is then enough toshow that C ϕ ( u ( A,S,m • ,f ) ◦ ) = u ( A,S,m • ,ϕ ◦ f ) ◦ . As we already know that both sides aremaps between the same objects, it suffices to prove this as maps in E M -SSet ,where it is indeed immediate from the definition that both sides are given by ( u ∗ × u ∗ ) − : E Inj(
S, ω ) × Q a ∈ A ∆ m a → E Inj( u ( S ) , ω ) × Q b ∈ u ( A ) ∆ m u − b ) . LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 21
Finally, it is obvious from the definition that C id = id, and that C ψ C ϕ = C ψϕ for any further E M -equivariant map ψ : Y → Z , which then completes the proofof the lemma. (cid:3) In order to construct the E M -equivariant refinement of the ‘last vertex map,’we need the following easy structural insight on the E M -simplicial sets appearingin the definition of C • : Remark . If A is any set and ( m a ) a ∈ A is an A -indexed family of non-negativeintegers, then Q a ∈ A ∆ m a is isomorphic to the nerve of the poset Q a ∈ A [ m a ]. Thelatter has a unique terminal object (i.e. maximum element) given by ( m a ) a ∈ A , andwe write ∗ for the corresponding vertex of Q a ∈ A ∆ m a , i.e. ∗ = Q a ∈ A ∆ { m a } .If S is any further set, then E Inj(
S, ω ) is by construction the nerve of a categoryin which there is precisely one morphism u → v for any two objects u, v . It follows,that there exists for any u ∈ Inj(
S, ω ) and any vertex x of E Inj(
S, ω ) × Q a ∈ A ∆ m a a unique edge x → ( u, ∗ ).Finally, again using that in E Inj(
S, ω ) and [ m a ] there is at most one morphism x → y for any two objects x, y , we see that any n -simplex of E Inj(
S, ω ) × Q a ∈ A ∆ m a is completely determined by its ( n + 1)-tuple of vertices. Conversely, such an( n + 1)-tuple ( x , . . . , x n ) comes from an n -simplex if and only if there exists foreach 1 ≤ i ≤ n a (necessarily unique) edge x i − → x i . Construction . Let X be an E M -simplicial set. We define ǫ : N( C X ) → X asfollows: if ( A , S , m (0) • , f ) α −→ ( A , S , m (1) • , f ) → · · · α k −−→ ( A k , S k , m ( k ) • , f k ) is a k -simplex of N( C X ), then we denote by σ α • the unique k -simplex of E Inj( S k , ω ) × Q a ∈ A k ∆ m ( k ) a whose ℓ -th vertex ( ℓ = 0 , . . . , k ) is given by α k · · · α ℓ +1 ( ι S ℓ , ∗ ), where ι S ℓ ∈ Inj( S ℓ , ω ) denotes the inclusion. This is indeed well-defined as there exists anedge α ℓ ( ι S ℓ − , ∗ ) → ( ι S ℓ , ∗ ) in E Inj( S ℓ , ω ) × ∆ n ℓ for all 1 ≤ ℓ ≤ n .We then set ǫ ( α • ) := f k ( σ α • ) ∈ X k . Proposition 3.12.
The above defines a natural transformation ǫ : N ◦ C • ⇒ id ofendofunctors of E M -SSet .Proof. Let us first show that ǫ X is indeed a simplicial map; this is completely anal-ogous to the argument for the usual last vertex map, but we nevertheless include itfor completeness. For this we let ( A , S , m (0) • , f ) α −→ · · · α k −−→ ( A k , S k , m ( k ) • , f k )be any k -simplex of N C X , and we let ϕ : [ ℓ ] → [ k ] be any map in ∆. Then ϕ ∗ ( σ α • ) is the unique ℓ -simplex of E Inj( S k , ω ) × Q a ∈ A k ∆ m ( k ) a with i -th vertex α k · · · α ϕ ( i )+1 ( ι S ϕ ( i ) , ∗ ). On the other hand, σ ϕ ∗ α • is by definition the unique ℓ -simplex of E Inj( S ϕ ( ℓ ) , ω ) × Q a ∈ A ∆ m ( ϕ ( ℓ )) a with i -th vertex given by ϕ ∗ ( α • ) ℓ · · · ϕ ∗ ( α • ) i +1 ( ι S ϕ ( i ) , ∗ ) = α ϕ ( ℓ ) · · · α ϕ ( i )+1 ( ι S ϕ ( i ) , ∗ ) . Thus, α k · · · α ϕ ( ℓ )+1 ( σ ϕ ∗ ( α • ) ) = ϕ ∗ σ α • and hence ǫ ( ϕ ∗ ( α • )) = f ϕ ( ℓ ) ( σ ϕ ∗ ( α • ) ) = f k α k · · · α ϕ ( ℓ )+1 ( σ ϕ ∗ ( α • ) )= f k ( ϕ ∗ σ α • ) = ϕ ∗ f k ( σ α • ) = ϕ ∗ ǫ ( α • ) , i.e. ǫ is indeed a simplicial map. Next, we have to show that ǫ is E M -equivariant, for which we let ( u , . . . , u k ) ∈M k +1 arbitrary. Then we have a commutative diagram( A , S , m (0) • , f ) ( A , S , m (1) • , f ) · · · ( A k , S k , m ( k ) • , f k ) u . ( A , S , m (0) • , f ) u . ( A , S , m (1) • , f ) · · · u k . ( A k , S k , m ( k ) • , f k ) α ( u ) ◦ ( u ) ◦ α α k ( u k ) ◦ in C X , where the lower row is given by ( u , . . . , u k ) .α • . Thus, σ ( u ,...,u k ) .α • is theunique k -simplex with i -th vertex given by(3.2) ( u k ) ◦ α k · · · α i +1 ( u i ) − ◦ ( ι u ( A i ) , ∗ ) . By definition, ( u i ) − ◦ ( ι u ( A i ) , ∗ ) = ( u i | A , ∗ ) = u i . ( ι A , ∗ ); E M -equivariance of α k , . . . ,α i +1 therefore implies that (3 .
2) equals ( u k ) ◦ ( u i . ( α k · · · α i +1 ( ι A i , ∗ ))). Comparingvertices, we conclude that σ ( u ,...,u k ) .α • = ( u ∗ k × u ∗ k ) − (( u , . . . , u k ) .σ α • ), hence ǫ (( u , . . . , u k ) .α • ) = f k ◦ ( u ∗ k × u ∗ k )( σ ( u ,...,u k ) .α • ) = f k (( u , . . . , u k ) .σ α • )= ( u , . . . , u k ) .f k ( σ α • ) = ( u , . . . , u k ) .ǫ ( α • ) , i.e. ǫ is E M -equivariant.Finally, let us show that ǫ is natural. If ϕ : X → Y is any E M -equivariant map,then N( C ϕ )( α • ) = ( A , S , m (0) • , ϕ ◦ f ) α −→ · · · α k −−→ ( A k , S k , m ( k ) , ϕ ◦ f k ). Thus, σ N( C ϕ )( α • ) = σ α • and ǫ (N( C ϕ )( α • )) = ϕf k ( σ N( C ϕ )( α • ) ) = ϕf k ( σ α • ) = ϕ ( ǫ ( α • )).This completes the proof of the proposition. (cid:3) Remark . Let C be a small E M -category. Then applying Construction 3.11 tothe E M -simplicial set N( C ) yields an E M -equivariant map N( C N C ) → N( C ). Asthe nerve is fully faithful, this is induced by a unique functor ˜ ǫ : C N C → C , whichis then automatically again E M -equivariant. This way, we get a (unique) naturaltransformation ˜ ǫ : C • ◦ N ⇒ id of endofunctors of E M -Cat with N(˜ ǫ C ) = ǫ N C .Explicitly, ˜ ǫ C is the functor sending an object ( A, S, m • , f ) to the object cor-responding to the image of ( ι S , ∗ ) under f , and a morphism α : ( A, S, m • , f ) → ( B, T, n • , g ) to the morphism corresponding to the image under g of the uniqueedge α ( ι A , ∗ ) → ( ι B , ∗ ) of E Inj(
T, ω ) × Q b ∈ B ∆ n b .So far we have only considered C • as a functor E M -SSet → E M -Cat τ .However, we can formally lift this to E M - G -SSet → E M - G -Cat τ by pullingthrough the G -action via functoriality. Explicitly, if X is an E M - G -simplicial set,then g ∈ G acts on ( A, S, m • , f ) via g. ( A, S, m • , f ) = ( A, S, m • , ( g. –) ◦ f ), and if α : ( A, S, m • , f ) → ( A ′ , S ′ , m ′• , f ′ ), then g.α is the same morphism of E M -simplicialsets, but this time considered as a map ( A, S, m • , ( g. –) ◦ f ) → ( A ′ , S ′ , m ′• , ( g. –) ◦ f ′ ).It follows formally that ǫ and ˜ ǫ are G -equivariant, and that they define naturaltransformations of endofunctors of E M - G -SSet and E M - G -Cat , respectively.4. An Unstable Comparison
In this section we will prove the following predecessor to Theorem B:
Theorem 4.1.
The nerve and the functor C • from Lemma 3.9 restrict to mutuallyinverse homotopy equivalences C • : E M - G -SSet τ ⇄ E M - G -Cat τ : N LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 23 with respect to the G -global weak equivalences on both sides. More precisely, thenatural transformations ǫ from Construction 3.11 and ˜ ǫ from Remark 3.13 restrictto natural levelwise G -global weak equivalences between the composites N ◦ C • and C • ◦ N and the respective identities. The proof will be given later in this section after some preparations.
Remark . Let H ⊂ M be any subgroup and let ϕ : H → G be a group homomor-phism. We will now make the ϕ -fixed points of C X explicit: if ( A, S, m • , f ) is anyobject, and h ∈ H , then ( h, ϕ ( h )) . ( A, S, m • , f ) = ( h ( A ) , h ( S ) , m h − ( • ) , ( ϕ ( h ) . –) ◦ f ◦ ( h ∗ × h ∗ )). In particular, the first three components are fixed if and only if A, S ⊂ ω are H -subsets, and m • is constant on H -orbits, i.e. m h.a = m a for all a ∈ A, h ∈ H . In this case, we have an H -action on E Inj(
S, ω ) × Q a ∈ A ∆ m a givenby h. – = ( h ∗ × h ∗ ) − , i.e. H acts by the diagonal of the H -action on S and the‘shuffling’ action on Q a ∈ A ∆ m a ; we call this the preaction as it is essentially givenby precomposition. The condition that ( ϕ ( h ) . –) ◦ f ◦ ( h ∗ × h ∗ ) = f for all h ∈ H then precisely means that f is an H -equivariant map(4.1) E Inj(
S, ω ) × Y a ∈ A ∆ m a → ϕ ∗ X with respect to the above H -action on the source. In analogy with the terminologyfor the action on the source, we will also refer to the H -action on the target as preaction .The E M -action on E Inj(
S, ω ) × Q a ∈ A ∆ m a commutes with the preaction, and inparticular restricting it to H gives another H -action commuting with the preaction,and that we denote by ‘ ∗ ’ instead of the usual ‘.’ in order to avoid confusion. Wewill refer to this action as the postaction as it is given by postcomposition. The pre-and postaction together make E Inj(
S, ω ) × Q a ∈ A ∆ m a into an ( H × H )-simplicialset. Similarly the E M -action on X gives rise to another H -action (again denotedby ∗ and again called the postaction ) commuting with the one given by restring the G -action along ϕ , making it into another ( H × H )-simplicial set. As f is always E M -equivariant, the above condition that (4 .
1) be H -equivariant with respect tothe preactions is then equivalent to f being ( H × H )-equivariant and equivalent to f being ∆-equivariant, where ∆ denotes the diagonal subgroup of H × H .Now let α : ( A, S, m • , f ) → ( A ′ , S ′ , m ′• , f ′ ) be a map of ϕ -fixed objects. Then( h, ϕ ( h )) .α is again a map ( A, S, m • , f ) → ( A ′ , S ′ , m ′• , f ′ ) for all h ∈ H , so α =( h, ϕ ( h )) .α as morphisms of C X if and only if both sides agree in E M -SSet . Butacting with ϕ ( h ) does not affect α as a morphism of E M -SSet , and the H -actionis given by conjugating with ( h ∗ × h ∗ ). Thus, α is a ϕ -fixed point if and only if it is H -equivariant with respect to the preactions constructed above. As before this isequivalent to α being ( H × H )-equivariant and equivalent to α being ∆-equivariant.The following technical lemma provides the necessary equivariant informationabout the above objects: Lemma 4.3.
Let S ⊂ ω be finite, let Y ∈ SSet be isomorphic to the nerve of aposet with a maximum element, and let ∗ denote the corresponding vertex of Y .Write X := E Inj(
S, ω ) × Y and let H be any group. (1) Any H -action on X restricts to an H -action on E Inj(
S, ω ) × {∗} . (2) If the H -action on X is through E M -equivariant maps, then its restrictionto E Inj(
S, ω ) × {∗} is induced by a unique H -action on S . (3) Assume again that H acts on X through E M -equivariant maps, so that X is an ( E M × H ) -simplicial set, but assume moreover that H is a subgroupof M . Let ∆ be the diagonal subgroup of M × H , let T be any H -subsetof ω , and consider E Inj(
S, T ) × Y with the restriction of the ∆ -action on E Inj(
S, ω ) × Y .Then (cid:0) E Inj(
S, T ) × Y (cid:1) ∆ is contractible provided that there exists an H -equivariant injection S → T with respect to the H -action on S from (2) .Proof. For the first statement we observe that X is canonically identified withthe nerve of C := E Inj(
S, ω ) × P , and as N is fully faithful, it suffices to provethe analogous statement for C . For this it is then enough to observe that anyisomorphism of categories preserves the full subcategory spanned by the terminalobjects, and that this is precisely given by E Inj(
S, ω ) × {∗} in our case.For the second statement we observe that evaluation at ι S provides a bijec-tion Hom E M ( E Inj(
S, ω ) , E Inj(
S, ω )) → ( E Inj(
S, ω ) [ S ] ) = Inj( S, ω ) [ S ] by Proposi-tion 2.15. On the other hand, we have a map(4.2) Σ S → Hom E M ( E Inj(
S, ω ) , E Inj(
S, ω ))sending σ to the map given by precomposition with σ − . The composition Σ S → Inj(
S, ω ) [ S ] is then given by σ ι S σ − , which is obviously bijective. We concludethat also (4 .
2) is bijective. In particular, there exists for each h ∈ H a unique σ ( h )such that h. – : E Inj(
S, ω ) → E Inj(
S, ω ) agrees with – ◦ σ ( h ) − . It only remains toshow that this defines an action on S , i.e. that σ is a group homomorphism, forwhich it is enough to observe that (4 .
2) is a monoid homomorphism.For the final statement, we again switch to the categorical perspective. As thenerve is continuous, it then suffices to show that C ∆ is contractible, for which itis enough that it has a terminal object. For this we observe that there is at mostone map x → y for any x, y ∈ C = E Inj(
S, ω ) × P . Thus, a morphism in C is fixedby ∆ if and only if its two endpoints are, i.e. C ∆ is a full subcategory of C . It istherefore enough to show that one of the terminal objects of C is fixed by ∆. Butby the previous steps, the ∆-action restricts to a ∆-action on E Inj(
S, T ) × {∗} ,where ( h, h ) for h ∈ H acts by ( f, ∗ ) ( h ◦ f ◦ ( h − . –) , ∗ ). Obviously, a terminalobject ( f, ∗ ) is fixed under this action if and only if the injection f is H -equivariant,and such an f exists by assumption. (cid:3) Proof of Theorem 4.1.
We will show that ǫ X is a G -global weak equivalence for eachtame E M - G -simplicial set X . If C is any tame E M - G -category, then applying thisto N( C ) will also show that N(˜ ǫ C ) = ǫ N C is a G -global weak equivalence, and as thenerve creates the G -global weak equivalences in E M - G -Cat τ , this will then implythat also ˜ ǫ is a levelwise G -global weak equivalence. Moreover, we can concludefrom this by 2-out-of-3 that N ◦ C • is homotopical, and hence so is C • , which thenaltogether implies the theorem.Therefore let us fix a tame E M - G -simplicial set X , a universal subgroup H ⊂M , and a homomorphism ϕ : H → G . We will show that the restriction of ǫ toN( C X ) [ T ] → X [ T ] induces a weak equivalence on ϕ -fixed points for each finite H -subset T ⊂ ω containing an H -fixed point. For varying T , these exhaust N C X and X , as both are tame by assumption and since any finite set T ′ ⊂ ω is containedin a finite H -subset containing an H -fixed point (the latter uses that ω H = ∅ by LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 25 universality). Passing to the filtered colimit over all such T will thus yield the claimas filtered colimits in SSet are homotopical and commute with finite limits.To prove the claim, we fix t ∈ T H and we consider the functor i : ∆ ↓ X ϕ [ T ] → ( C X ) [ T ] sending an object k : ∆ n → X to ( { t } , T, n, ˜ k ), where ˜ k is the unique E M -equivariant map E Inj(
T, ω ) × ∆ n → X with ˜ k ( ι T , –) = k , and a morphism α : k → ℓ to E Inj(
T, ω ) × α . We omit the easy verification that i is well-defined.We now claim that i actually lands in the ϕ -fixed points. Let us first checkthis on objects: if k : ∆ n → X is any object of ∆ ↓ X ϕ [ T ] , then we have to showthat ( { t } , T, n, ˜ k ) is ϕ -fixed. But indeed, T ⊂ ω is an H -subset by assumption, { t } ⊂ ω is an H -subset as t ∈ T H , and any family on { t } is constant on orbits fortrivial reasons, so it only remains to show by Remark 4.2 that ˜ k is equivariant withrespect to the preactions. But since for any h ∈ H both ˜ k ◦ ( h. –) and ( h. –) ◦ ˜ k are E M -equivariant, it suffices to show that they agree on { ι T } × ∆ n , for which we let σ denote any simplex of ∆ n . Then˜ k ( h.ι T , σ ) = ˜ k ( ι T ◦ h − | T , σ ) = ˜ k ( h − ∗ ι T , σ ) = h − ∗ (˜ k ( ι T , σ )) = ϕ ( h ) . (˜ k ( ι T , σ ))as desired, where the last equation uses that ˜ k ( ι T , σ ) = k ( σ ) is ϕ -fixed.Now let ℓ : ∆ n ′ → X ϕ [ T ] be another object of ∆ ↓ X ϕ [ T ] and let α : k → ℓ be anymorphism. Then E Inj(
T, ω ) × α : E Inj(
T, ω ) × ∆ n → E Inj(
T, ω ) × ∆ n ′ is obviouslyequivariant in the preactions, hence ϕ -fixed by Remark 4.2. This completes theproof that i lands in ( C X ) ϕ [ T ] .Next, we consider the composite(4.3) N(∆ ↓ X ϕ [ T ] ) N( i ) −−−→ N (cid:0) ( C X ) ϕ [ T ] (cid:1) ∼ = (N C X ) ϕ [ T ] ǫ ϕ [ T ] −−→ X ϕ [ T ] , where the unlabelled isomorphism comes from Example 2.7 together with the factthat N is a right adjoint.If g α −→ g → · · · α k −−→ g k is a k -simplex of the left hand side (where each f ℓ isa map ∆ n ℓ → X ϕ [ T ] ), then the above composite sends this to the image of σ under˜ g k , where σ is the unique ℓ -simplex of E Inj(
T, ω ) × ∆ m k whose ℓ -th vertex is i ( α k ) · · · i ( α ℓ +1 )( ι T , ∗ ) = ( E Inj(
T, ω ) × α k ) · · · ( E Inj(
T, ω ) × α ℓ +1 )( ι T , ∗ )= ( ι T , α k · · · α ℓ +1 ( ∗ )) = ( ι T , α k · · · α ℓ +1 (∆ { n ℓ } ))Thus, if τ is the unique k -simplex of ∆ n k with ℓ -th vertex α k · · · α ℓ +1 (∆ { n ℓ } ), then σ = ( ι T , τ ), and hence ˜ g k ( σ ) = ˜ g k ( ι T , τ ) = g k ( τ ) = ǫ ( α • ). We therefore concludethat the composite (4 .
3) agrees with the last vertex map N(∆ ↓ X ϕ [ T ] ) → X ϕ [ T ] , sothat it is a weak homotopy equivalence by Proposition 3.3.It is therefore enough to show that i is a weak homotopy equivalence. ByQuillen’s Theorem A [Qui73, § i ↓ ( A, S, m • , f )has weakly contractible nerve for each ( A, S, m • , f ) ∈ ( C X ) ϕ [ T ] .So let ( A, S, m • , f ) be any ϕ -fixed point supported on T . Then K := E Inj(
S, T ) × Q a ∈ A ∆ m a is canonically identified with the subcomplex of E Inj(
S, ω ) × Q a ∈ A ∆ m a consting of the simplices supported on T , and from this it inherits the two commut-ing H -actions considered before: the postaction given by restriction of the E M -action on E Inj(
S, ω ) (i.e. induced by the H -action on T ) and the preaction givenby the H -actions on A and S . We will be interested in the fixed points K ∆ for thediagonal of these two actions. Namely, let us define a functor c : i ↓ ( A, S, m • , f ) → ∆ ↓ K ∆ as follows: an object of the left hand side consists by definition of a map g : ∆ n → X ϕ [ T ] together with a ϕ -fixed morphism α : i ( g ) → ( A, S, m • , f ), i.e. an( E M × H )-equivariant map α : E Inj(
T, ω ) × ∆ n → E Inj(
S, ω ) × Q a ∈ A ∆ m a suchthat ˜ g = f α . We now claim that the composition∆ n ( ι T , –) −−−−→ E Inj(
T, ω ) × ∆ n α −→ E Inj(
S, ω ) × Y a ∈ A ∆ m a actually lands in K ∆ . Indeed, it is clear that it lands in K , so we only have to showthat α ( ι T , σ ) is ∆-fixed for each simplex σ of ∆ n . But indeed, as α is ∆-equivariant,it suffices that ι T is a ∆-fixed point of E Inj(
T, ω ), which is immediate from thedefinition. With this established, we now define c ( g, α ) as α ( ι T , –) considered as amap ∆ n → K ∆ .If ( g ′ : ∆ n ′ → X ϕ [ T ] , α ′ : i ( g ′ ) → ( A, S, m • , f )) is another object of i ↓ ( A, S, m • , f ),then a morphism ( g, α ) → ( g ′ , α ′ ) is given by a map a : ∆ n → ∆ n ′ such that g = g ′ ◦ a (i.e. a is a map g → g ′ in ∆ ↓ X ϕ [ T ] ) and α = i ( a ) ◦ α ′ . As i ( a ) = E Inj(
T, ω ) × a ,restricting to { ι T } × ∆ n shows that α ( ι T , –) = α ′ ( ι T , –) ◦ a , i.e. a also defines amorphism c ( g, α ) → c ( g ′ , α ′ ) in ∆ ↓ K ∆ , which we take as the definition of c ( a ). Itis clear that c is a functor. Claim. c is an equivalence of categories. Proof.
We will show that c is fully faithful and surjective on objects; in fact, itis not hard to show that c is also injective on objects, hence an isomorphism ofcategories, but we will not need this.It is clear from the definition that c is faithful. To see that it is full we let( g : ∆ n → X ϕ [ T ] , α ) , ( g ′ : ∆ n ′ → X ϕ [ T ] , α ′ ) be objects of the left hand side, and we let a : ∆ n → ∆ n ′ be a morphism c ( g, α ) → c ( g ′ , α ′ ), i.e.(4.4) α ( ι T , –) = α ′ ( ι T , –) ◦ a . We want to show that a also defines a morphism ( g, α ) → ( g ′ , α ′ ), i.e. that the twotriangles∆ n ∆ n ′ X ϕ [ T ] a g g ′ and E Inj(
T, ω ) × ∆ n E Inj(
T, ω ) × ∆ n ′ E Inj(
S, ω ) × Q a ∈ A ∆ m a E Inj(
T,ω ) × a α α ′ commute. For the second one we observe that both paths through the diagramare E M -equivariant, so that it suffices to show this after restring to { ι T } × ∆ n ,where this is precisely the identity (4 . g = f ◦ α as α is a morphism i ( g ) → ( A, S, m • , f ), hence g = f ◦ α ( ι T , –) and analogously g ′ = f ◦ α ′ ( ι T , –).Thus, a also defines a morphism ( g, α ) → ( g ′ , α ′ ) which is then obviously thedesired preimage.Finally, let us show that c is surjective on objects. We let ˆ α : ∆ n → K ∆ be anymap; then the composition∆ n ˆ α −→ K ∆ = E Inj(
S, T ) × Y a ∈ A ∆ m a ! ∆ ֒ → E Inj(
S, ω ) × Y a ∈ A ∆ m a LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 27 by construction lands in the subcomplex of those simplices that are supported on T ,so it extends to a unique E M -equivariant map α : E Inj(
T, ω ) × ∆ n → E Inj(
S, ω ) × Q a ∈ A ∆ m a .We claim that ( f ˆ α, α ) defines an element of i ↓ ( A, S, m • , f ), which amountsto saying that f ˆ α : ∆ n → X factors through X ϕ [ T ] , that α is H -equivariant withrespect to the preactions, and that the diagram E Inj(
T, ω ) × ∆ n E Inj(
S, ω ) × Q a ∈ A ∆ m a X α ] ( f ˆ α ) f commutes. For the first statement, we observe that f ˆ α lands in X [ T ] as ˆ α lands in( E Inj(
S, ω ) × Q a ∈ A ∆ m a ) [ T ] and because f is E M -equivariant. To see that it alsolands in the ϕ -fixed points, it suffices to observe that f restricts to ( E Inj(
S, ω ) × Q a ∈ A ∆ m a ) ∆ → X ϕ by Remark 4.2.For the second statement it is again enough that h. (ˆ α ( σ )) = α ( h.ι T , σ ) for allsimplices σ of ∆ n and all h ∈ H . But indeed, h.ι T = h − ∗ ι T as before, so α ( h.ι T , σ ) = h − ∗ (ˆ α ( σ )), and this in turn agrees with h. (ˆ α ( σ )) because ˆ α ( σ ) is∆-fixed.Finally, for the third statement it suffices again to check this on { ι T }× ∆ n , whereit holds tautologically. Altogether we have shown that ( f ˆ α, α ) defines an elementof i ↓ ( A, S, m • , f ). It is then immediate from the definition that c ( f ˆ α, α ) = ˆ α ,which completes the proof of the claim. △ We conclude that in particular N( i ↓ ( A, S, m • , f )) ≃ N(∆ ↓ K ∆ ). By Propo-sition 3.3 we further see that N(∆ ↓ K ∆ ) is weakly equivalent to K ∆ , so it onlyremains to prove that the latter is (weakly) contractible. This is a direct appli-cation of Lemma 4.3: the restriction of the preaction on E Inj(
S, ω ) × Q a ∈ A ∆ m a to E Inj(
S, ω ) × {∗} is by construction induced by the H -action on S coming fromthe H -action on ω . On the other hand, S is a subset of supp( A, S, m • , f ) ⊂ T by Lemma 3.7, so the inclusion S ֒ → T is the desired H -equivariant injection.Altogether, this completes the proof of the theorem. (cid:3) So far we have only used the third part of Lemma 4.3. However, we will nowneed its full strength for the proof of the following result:
Proposition 4.4.
Let X be a tame E M - G -simplicial set. Then C X is weaklysaturated. Before we can prove the proposition, we need to understand the categories C ‘ h ’ ϕX better: Remark . Let us first assume that X = ∗ , and let Φ ∈ C ‘ h ’ ϕ ∗ arbitrary. As thereare no non-trivial actions on ∗ , this means that Φ : EH → C ∗ is H -equivariant withrespect to the restriction of the M -action on C ∗ to H .Let us write ( A, S, m • , ∗ ) := Φ(1) (where ∗ will always denote the unique mapfrom an implicitly understood object to the fixed terminal simplicial set ∗ ). Thenwe have for each h ∈ H an E M -equivariant self-map h. – of E Inj(
S, ω ) × Q a ∈ A ∆ m a given by the compositionΦ(1) h ◦ =( h ∗ × h ∗ ) − −−−−−−−−−−→ h. Φ(1) = Φ( h ) Φ(1 ,h ) −−−−→ Φ(1); it is not hard to check that this defines an H -action, also cf. [Sch19b, Construc-tion 7.4]. In analogy with Remark 4.2 we call this the preaction induced by Φ.Now let Ψ be any other element of C ∗ , Ψ(1) =: ( B, T, n • , ∗ ), and let α : Φ → Ψbe any morphism. Then α h = h.α , so α is completely determined by the E M -equivariant map α : E Inj(
S, ω ) × Q a ∈ A ∆ m a → E Inj(
T, ω ) × Q b ∈ B ∆ n b (assumingwe have fixed the objects Φ and Ψ). In the diagram(4.5) Φ(1) h. Φ(1) Φ( h ) Φ(1)Ψ(1) h. Ψ(1) Ψ( h ) Ψ(1) α h ◦ h.α α h Φ(1 ,h ) α h ◦ Ψ(1 ,h ) the left hand square commutes by naturality of h ◦ , the middle square commutes byequivariance of α , and the right hand square commutes by naturality of α . Thus,the total rectangle commutes, i.e. α is equivariant in the preactions.Conversely, if α : Φ(1) → Ψ(1) is H -equivariant, let us define α h := h.α =( h ∗ × h ∗ ) − ◦ α ◦ ( h ∗ × h ∗ ). Then the outer rectangle in (4 .
5) commutes, and sodo the left hand and middle square by the same arguments as above. As all hor-izontal morphisms are isomorphisms, we conclude that also the right hand squarecommutes, i.e. α is compatible with the edges (1 , h ) in EH . Since these gener-ate EH as a groupoid, we conclude that α is natural. As it is H -equivariant byconstruction, it is therefore a morphism Φ → Ψ in C ∗ . Altogether we have shownthat the assigment α α defines a bijection between Hom(Φ , Ψ) and the set of( E M × H )-equivariant maps E Inj(
S, ω ) × Q a ∈ A ∆ m a → E Inj(
T, ω ) × Q b ∈ B ∆ n b .Now assume X ∈ E M - G -SSet is arbitrary, and let Φ ∈ C ‘ h ’ ϕX . Applying thefunctor C ‘ h ’ ϕX → C ‘ h ’ ϕ ∗ induced by the unique map X → ∗ to Φ then shows that thecomposites h. – : E Inj(
S, ω ) × Y a ∈ A ∆ m a ( h ∗ × h ∗ ) − −−−−−−−→ E Inj( h ( S ) , ω ) Y b ∈ h ( A ) ∆ m h − b ) Φ(1 ,h ) −−−−→ E Inj(
S, ω ) × Y a ∈ A ∆ m a define an H -action on E Inj(
S, ω ) × Q a ∈ A ∆ m a by E M -equivariant maps, which weagain call the preaction induced by Φ. However, the h. – typically do not define self-maps of Φ(1) anymore; instead, the identities f ◦ Φ(1 , h ) = ( ϕ ( h ) . –) ◦ f ◦ ( h ∗ × h ∗ )coming from the requirement that Φ(1 , h ) be a map ( h, ϕ ( h )) . Φ(1) = Φ( h ) → Φ(1)imply that f : E Inj(
S, ω ) × Q a ∈ A ∆ m a → ϕ ∗ X is H -equivariant with respect to theabove H -action on the source, i.e. f is equivariant in the preactions. As before,this is equivalent for the E M -equivariant map f to being equivariant with respectto the diagonal actions.Next, let Ψ ∈ C ‘ h ’ ϕX be another object, and let α : Φ → Ψ be any morphism. Asbefore, α is completely determined by α , and pushing α forward to C ∗ shows that α is equivariant with respect to the preactions. In addition, the diagram(4.6) E Inj(
S, ω ) × Q a ∈ A ∆ m a E Inj(
T, ω ) × Q b ∈ B ∆ n b X f α g LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 29 commutes as α is a morphism Φ(1) → Ψ(1).On the other hand, if α is an ( E M× H )-equivariant map making (4 .
6) commute,then the above shows that α h := ( h ∗ × h ∗ ) − ◦ α ◦ ( h ∗ × h ∗ ) defines an H -equivariantnatural transformation, so to see that this defines a map Φ → Ψ in C ‘ h ’ ϕX it onlyremains to show that α h is a map Φ( h ) → Ψ( h ), i.e. that (cid:0) ( ϕ ( h ) . –) ◦ g ◦ ( h ∗ × h ∗ ) (cid:1) ◦ α h = ( ϕ ( h ) . –) ◦ f ◦ ( h ∗ × h ∗ ). This however follows immediately from the explicitdescription of α h .Altogether we have shown that α α defines a bijection between Hom(Φ , Ψ)and the set of those ( E M × H )-equivariant maps E Inj(
S, ω ) × Q a ∈ A ∆ m a → E Inj(
T, ω ) × Q b ∈ B ∆ n b making the diagram (4 .
6) commute. As before, ( E M × H )-equivariance is equivalent to being E M - and ∆-equivariant. Proof of Proposition 4.4.
Let H ⊂ M be a universal subgroup and let ϕ : H → G be a group homomorphism. We have to show that the canonical map c : C ϕX → C ‘ h ’ ϕX is a weak homotopy equivalence. To this end we consider for each (fi-nite) H -subset T ⊂ ω the following full subcategory ( C ‘ h ’ ϕX ) h T i ⊂ C ‘ h ’ ϕX : if Φ ∈ C ‘ h ’ ϕX , ( A, S, m • , f ) := Φ(1), then Remark 4.5 describes a canonical H -action on E Inj(
S, ω ) × Q a ∈ A ∆ m a , whose restriction to E Inj(
S, ω ) ×{∗} is induced by a unique H -action on S according to Lemma 4.3; we now declare that Φ should belong to( C ‘ h ’ ϕX ) h T i if and only if the H -set S admits an H -equivariant injection into T .If ( A, S, m • , f ) is ϕ -fixed, then A, S ⊂ ω are H -subsets, and the above H -actionon E Inj(
S, ω ) × Q a ∈ A ∆ m a is simply the preaction from Remark 4.2. In particular,its restriction to E Inj(
S, ω ) × {∗} is induced by the tautological H -action on S ⊂ ω .Thus, if ( A, S, m • , f ) is supported on T , then the inclusion S ֒ → T is H -equivariantwith respect to the above action, so that c : C ϕX → C ‘ h ’ ϕX restricts to ( C X ) ϕ [ T ] → ( C ‘ h ’ ϕX ) h T i .Next, we observe that the ( C ‘ h ’ ϕX ) h T i exhaust C ‘ h ’ ϕX when we let T run through allfinite H -subsets of ω with T H = ∅ : indeed, if Φ is arbitrary, ( A, S, m • , f ) := Φ(1),then we consider the finite H -set S ∐ {∗} where H acts on S as above and triviallyon ∗ . As ω is a universal H -set, there exists an H -equivariant injection S ∐{∗} ω ,whose image is then the desired T . Thus, the inclusions express C ‘ h ’ ϕX as a filteredcolimit along the inclusions of the ( C ‘ h ’ ϕX ) h T i over all finite H -subsets T ⊂ ω with T H = ∅ .Altogether, we are reduced to showing that ( C X ) ϕ [ T ] → ( C ‘ h ’ ϕX ) h T i is a weakhomotopy equivalence for all such T , for which it is enough by 2-out-of-3 that thecomposition j : ∆ ↓ X ϕ [ T ] i −→ ( C X ) ϕ [ T ] c −→ ( C ‘ h ’ ϕX ) h T i is a weak homotopy equivalence, where i is the weak homotopy equivalence fromthe proof of Theorem 4.1.For this it is again enough by Quillen’s Theorem A that the slice j ↓ Φ has weaklycontractible nerve for each Φ ∈ ( C ‘ h ’ ϕX ) h T i . To prove this, let Φ(1) =: ( A, S, m • , f )and define K := E Inj(
S, T ) × Q a ∈ A ∆ m a with H -action via the H -action on T andthe restriction of the preaction on E Inj(
S, ω ) × Q a ∈ A ∆ m a induced by Φ. UsingRemark 4.5 one can show precisely as in the proof of Theorem 4.1 that we havean equivalence of categories d : j ↓ Φ → ∆ ↓ K ∆ sending α : j ( g : ∆ n → X ϕ [ T ] ) → Φ to α ( ι T , –) and a map ( g, α ) → ( g ′ , α ′ ) given by a : ∆ n → ∆ n ′ to the map d ( g, α ) → d ( g ′ , α ′ ) given by the same a . In particular, we conclude together withProposition 3.3 that N( j ↓ Φ) ≃ N(∆ ↓ K ∆ ) ≃ K ∆ . By definition of ( C ‘ h ’ ϕX ) h T i there exists an H -equivariant injection S → T with respect to the H -action on S induced by Φ. Thus, Lemma 4.3 implies that K ∆ is contractible, which completesthe proof of the proposition. (cid:3) Remark . Let X = ∅ be a tame E M - G -simplicial set. Then C X is not saturated:Indeed, let x ∈ X be arbitrary and write S := supp( x ). Then there exists a(unique) E M -equivariant map ˜ x : E Inj(
S, ω ) → X sending ι S to x . We pick a finiteset T ⊂ ω r S with at least two elements and we define f as the composition E Inj( S ∪ T, ω ) × ∆ −→ E Inj( S ∪ T, ω ) res −−→ E Inj( S ) ˜ x −→ X. We moreover choose a universal subgroup H of M together with an isomorphism ψ : H → Σ T , and we write ϕ : H → h ∈ H we define its action on E Inj( S ∪ T, ω ) × ∆ as the unique self-map τ h sending ( u, u,
0) and ( u,
1) to ( u ◦ ( S ∪ ψ ( h − )) ,
1) for each u ∈ M . We omit the easyverification that this is a well-defined H -action.Let now a ∈ ω be any H -fixed point. It is then not hard to check thatΦ : EH → C X with Φ( h ) = ( { a } , h ( S ∪ T ) , , f ◦ ( h ∗ × h ∗ )) and structure mapsΦ( h , h ) = ( h ) ◦ τ h − h ( h ) − ◦ defines an element of C ‘ h ’ ϕX . The induced H -actionon E Inj( S ∪ T, ω ) × ∆ is then simply the one given above. By the description of themorphisms in C ‘ h ’ ϕX given in Remark 4.5 it is then enough to show that this is not H -equivariantly isomorphic to a simplicial set of the form E Inj(
U, ω ) × Q b ∈ B ∆ n b for some finite H -subsets B, U ⊂ ω , with H acting via its tautological actions on B and U .Indeed, if there were such an isomorphism α , then it would restrict to H -equivariant isomorphisms E Inj(
U, ω ) × Q b ∈ B ∆ { } ∼ = E Inj( S ∪ T, ω ) × ∆ { } and E Inj(
U, ω ) × Q b ∈ B ∆ { n b } ∼ = E Inj( S ∪ T, ω ) × ∆ { } . In particular, the two H -simplicial sets E Inj( S ∪ T, ω ) × ∆ { } and E Inj( S ∪ T, ω ) × ∆ { } would be H -equivariantly isomorphic. But this is obviously not the case as precisely one ofthem has trivial H -action, yielding the desired contradiction.With Proposition 4.4 at hand we can now prove: Corollary 4.7.
All the inclusions in E M - G -Cat τ,s ֒ → E M - G -Cat τ,ws ֒ → E M - G -Cat τ are homotopy equivalences with respect to the G -global weak equivalences.Proof. We already know this for the left hand inclusion by Corollary 1.31, so itsuffices to consider the right hand inclusion. We claim that C • ◦ N defines a ho-motopy inverse. Indeed, this lands in E M - G -Cat τ,ws by Proposition 4.4; more-over, the natural map ˜ ǫ : C N C → C is a G -global weak equivalence for every tame E M - G -category C by Theorem 4.1. As E M - G -Cat τ,ws is a full subcategory of E M - G -Cat τ , this immediately implies the claim. (cid:3) Lifting the Parsummable Structure
In this section we will prove the parsummable analogues of Theorem 4.1 andCorollary 4.7 by lifting C • to a functor G -ParSumSSet → G -ParSumCat and LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 31 showing that the natural transformations ǫ, ˜ ǫ are compatible with the resultingstructure. Construction . Let A ′ , S ′ be (finite) sets and let A ⊂ A ′ , S ⊂ S ′ . Let moreover( m a ) a ∈ A ′ be a family of non-negative integers. Then we define ρ A ′ ,S ′ A,S : E Inj( S ′ , ω ) × Y a ∈ A ′ ∆ m a → E Inj(
S, ω ) × Y a ∈ A ∆ m a as the product of the restriction E Inj( S ′ , ω ) → E Inj(
S, ω ) and the projection Q a ∈ A ′ ∆ m a → Q a ∈ A ∆ m a . Lemma 5.2.
Throughout, let m • be an appropriately indexed family of non-negativeintegers. (1) ρ A ′ ,S ′ A,S is E M -equivariant for all A ⊂ A ′ , S ⊂ S ′ . (2) If A ⊂ A ′ ⊂ A ′′ , S ⊂ S ′ ⊂ S ′′ , then ρ A ′ ,S ′ A,S ρ A ′′ ,S ′′ A ′ ,S ′ = ρ A ′′ ,S ′′ A,S . (3) ρ A,SA,S = id for all
A, S (4) If A, B are disjoint, and
S, T are disjoint, then ( ρ A ∪ B,S ∪ TA,S , ρ A ∪ B,S ∪ TB,T ) re-stricts to an isomorphism E Inj( S ∪ T, ω ) × Y i ∈ A ∪ B ∆ m i ∼ = E Inj(
S, ω ) × Y a ∈ A ∆ m a ! ⊠ E Inj(
T, ω ) × Y b ∈ B ∆ m b ! . (5) If A ′ , S ′ ⊂ ω , and u ∈ M , then ρ A ′ ,S ′ A,S ◦ ( u ∗ × u ∗ ) = ( u ∗ × u ∗ ) ◦ ρ u ( A ′ ) ,u ( S ′ ) u ( A ) ,u ( S ) for all A ⊂ A ′ , S ⊂ S ′ .Proof. The first three statements are obvious. For the fourth statement let usfirst show that ( ρ A ∪ B,S ∪ TA,S , ρ A ∪ B,S ∪ TB,T ) lands in the box product. Indeed, it sendsan n -simplex ( u , . . . , u n , ( σ i ) i ∈ A ∪ B ) to (cid:0) ( u | S , . . . , u n | S ; σ | A ) , ( u | T , . . . , u n | T ; σ | B ).Obviously, supp k ( u | S , . . . , u n | S ; σ | A ) = u k ( S ) and supp k ( u | T , . . . , u n | T ; σ | B ) = u k ( T ); as u k is injective and S ∩ T = ∅ , these are disjoint, i.e. this is indeed an n -simplex of the box product.Conversely, given any n -simplex (cid:0) ( u , . . . , u n ; σ ) , ( v , . . . , v n ; τ ) (cid:1) of the box prod-uct, ( u ∪ v , . . . , u n ∪ v n ; σ ∪ τ ) with( u k ∪ v k )( x ) = ( u k ( x ) if x ∈ Sv k ( x ) if x ∈ T and ( σ ∪ τ ) i = ( σ i if i ∈ Aτ i if i ∈ B is well-defined because S ∩ T = ∅ and A ∩ B = ∅ , respectively. Moreover, this is an n -simplex of the left hand side: u k ∪ v k is injective, since its restrictions to S and T are, and since ( u k ∪ v k )( S ) = u k ( S ) is disjoint from ( u k ∪ v k )( T ) = v k ( T ) by thesame support calculation as above. It is then trivial to check that this is inverse tothe restriction of ( ρ A ∪ B,S ∪ TA,S , ρ A ∪ B,S ∪ TB,T ), which completes the proof of the fourthstatement.For the final statement, it suffices to observe that the diagram E Inj( u ( S ′ ) , ω ) E Inj( S ′ , ω ) E Inj( u ( S ) , ω ) E Inj(
S, ω ) u ∗ res res u ∗ commutes as both paths through it are given by restricting along S → u ( S ′ ) , s u ( s ), and that Q t ∈ u ( S ′ ) ∆ m u − t ) Q s ∈ S ′ ∆ m s Q t ∈ u ( S ) ∆ m u − t ) Q s ∈ S ∆ m s u ∗ pr pr u ∗ commutes because after postcomposition with pr s , s ∈ S , both paths agree withthe projection Q t ∈ u ( S ′ ) ∆ m u − t ) → ∆ m s onto the u ( s )-th factor. (cid:3) Construction . Let
X, Y ∈ E M -SSet τ . We define ψ : C X ⊠ C Y → C X ⊠ Y as follows: an object (cid:0) ( A, S, m • , f ) , ( B, T, n • , g ) (cid:1) is sent to ( A ∪ B, S ∪ T, ( m ∪ n ) • , f ∪ g ) where ( m ∪ n ) i = ( m i if i ∈ An i if i ∈ B and f ∪ g = ( f ⊠ g ) ◦ ( ρ A ∪ B,S ∪ TA,S , ρ A ∪ B,S ∪ TB,T ) = ( f ◦ ρ A ∪ B,S ∪ TA,S , g ◦ ρ A ∪ B,S ∪ TB,T ). Moreover,a morphism (cid:0) ( A, S, m • , f ) , ( B, T, n • , g ) (cid:1) → (cid:0) ( A ′ , S ′ , m ′• , f ′ ) , ( B ′ , T ′ , n ′• , g ′ ) (cid:1) givenby a pair α : E Inj(
S, ω ) × Y a ∈ A ∆ m a → E Inj( S ′ , ω ) × Y a ′ ∈ A ′ ∆ m ′ a ′ β : E Inj(
T, ω ) × Y b ∈ B ∆ n b → E Inj( T ′ , ω ) × Y b ′ ∈ B ′ ∆ n ′ b ′ is sent to the morphism ( A ∪ B, S ∪ T, ( m ∪ n ) • , f ∪ g ) → ( A ′ ∪ B ′ , S ′ ∪ T ′ , ( m ′ ∪ n ′ ) • , f ′ ∪ g ′ ) given by the composition( ρ A ′ ∪ B ′ ,S ′ ∪ T ′ A ′ ,S ′ , ρ A ′ ∪ B ′ ,S ′ ∪ T ′ B ′ ,T ′ ) − ◦ ( α ⊠ β ) ◦ ( ρ A ∪ B,S ∪ TA,S , ρ A ∪ B,S ∪ TB,T )= ( ρ A ′ ∪ B ′ ,S ′ ∪ T ′ A ′ ,S ′ , ρ A ′ ∪ B ′ ,S ′ ∪ T ′ B ′ ,T ′ ) − ◦ ( αρ A ∪ B,S ∪ TA,S , βρ A ∪ B,S ∪ TB,T ) . Finally, we define ι : ∗ → C ∗ as the functor sending the unique object of the lefthand side to ( ∅ , ∅ , ∅ , E Inj( ∅ , ω ) × Q ∅ → ∗ . Proposition 5.4.
The above functors are well-defined and E M -equivariant.Proof. Let us first show that ψ is well-defined. For this we observe that A ∪ S isdisjoint from B ∪ T by Lemma 3.7, so in particular A ∩ B = ∅ . Thus, m ∪ n is well-defined. Moreover, ρ A ∪ B,S ∪ TA,S and ρ A ∪ B,S ∪ TB,T are E M -equivariant by Lemma 5.2-(1),so f ∪ g = ( f ⊠ g ) ◦ ( ρ A ∪ B,S ∪ TA,S , ρ A ∪ B,S ∪ TB,T ) is again E M -equivariant. This showsthat ψ is well-defined on objects. To prove that it is well-defined on morphisms, weobserve that as above A ′ ∩ B ′ = ∅ , S ′ ∩ T ′ = ∅ , so that ( ρ A ′ ∪ B ′ ,S ′ ∪ T ′ A ′ ,S ′ , ρ A ′ ∪ B ′ ,S ′ ∪ T ′ B ′ ,T ′ )is indeed invertible by Lemma 5.2-(4). By another application of Lemma 5.2-(1)we then see that ψ ( α, β ) is E M -equivariant. Finally,( f ′ ∪ g ′ ) ψ ( α, β ) = ( f ′ ⊠ g ′ )( α ⊠ β )( ρ A ∪ B,S ∪ TA,S , ρ A ∪ B,S ∪ TB,T )= (cid:0) ( f ′ α ) ⊠ ( g ′ β ) (cid:1) ( ρ A ∪ B,S ∪ TA,S , ρ A ∪ B,S ∪ TB,T )= ( f ⊠ g )( ρ A ∪ B,S ∪ TA,S , ρ A ∪ B,S ∪ TB,T ) = f ∪ g, LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 33 i.e. ψ ( α, β ) is indeed a morphism in C X ⊠ Y from ψ (( A, S, m • , f ) , ( B, T, n • , g )) to ψ (( A ′ , S ′ , m ′• , f ′ ) , ( B ′ , T ′ , n ′• , g ′ )).It is trivial to check that ψ is a functor. Let us now prove that it is E M -equivariant, for which we let u ∈ M be arbitrary. Then ψ ( u. ( A, S, m • , f ) , u. ( B, T, n • , g ))= ψ (cid:0) ( u ( A ) , u ( S ) , m u − ( • ) , f ◦ ( u ∗ × u ∗ )) , ( u ( B ) , u ( T ) , n u − ( • ) , g ◦ ( u ∗ × u ∗ )) (cid:1) = (cid:0) u ( A ∪ B ) , u ( S ∪ T ) , ( m u − ( • ) ∪ n u − ( • ) ) • , ( f ◦ ( u ∗ × u ∗ )) ∪ ( g ◦ ( u ∗ × u ∗ )) (cid:1) . It is clear that ( m u − ( • ) ∪ n u − ( • ) ) • = ( m ∪ n ) u − ( • ) , so for M -equivariance on objectsit only remains to show that ( f ◦ ( u ∗ × u ∗ )) ∪ ( g ◦ ( u ∗ × u ∗ )) = ( f ∪ g ) ◦ ( u ∗ × u ∗ ).But indeed, Lemma 5.2-(5) implies that(5.1) ( ρ A ∪ B,S ∪ TA,S , ρ A ∪ B,S ∪ TB,T ) ◦ ( u ∗ × u ∗ )= ( ρ A ∪ B,S ∪ TA,S ◦ ( u ∗ × u ∗ ) , ( ρ A ∪ B,S ∪ TB,T ◦ ( u ∗ × u ∗ ))= (( u ∗ × u ∗ ) ◦ ρ u ( A ∪ B ) ,u ( S ∪ T ) u ( A ) ,u ( S ) , ( u ∗ × u ∗ ) ◦ ρ u ( A ∪ B ) ,u ( S ∪ T ) u ( B ) ,u ( T ) ) , hence ( f ◦ ( u ∗ × u ∗ )) ∪ ( g ◦ ( u ∗ × u ∗ ))= ( f ◦ ( u ∗ × u ∗ ) ◦ ρ u ( A ∪ B ) ,u ( S ∪ T ) u ( A ) ,u ( S ) , g ◦ ( u ∗ × u ∗ ) ◦ ρ u ( A ∪ B ) ,u ( S ∪ T ) u ( B ) ,u ( T ) )= ( f ⊠ g ) ◦ (( u ∗ × u ∗ ) ◦ ρ u ( A ∪ B ) ,u ( S ∪ T ) u ( A ) ,u ( S ) , ( u ∗ × u ∗ ) ◦ ρ u ( A ∪ B ) ,u ( S ∪ T ) u ( B ) ,u ( T ) )= ( f ⊠ g ) ◦ ( ρ A ∪ B,S ∪ TA,S , ρ A ∪ B,S ∪ TB,T ) ◦ ( u ∗ × u ∗ )= ( f ∪ g ) ◦ ( u ∗ × u ∗ ) . Next, we have to show that ψ ( u ( A,S,m • ,f ) ◦ , u ( B,T,n • ,g ) ◦ ) = u ψ (( A,S,m • ,f ) , ( B,T,n • ,g )) ◦ .As we already know that both sides are maps between the same two objects in C X ⊠ Y , it suffices to show this as maps in E M -SSet , for which it is in turn enoughthat their inverses agree. But indeed, ψ ( u ( A,S,m • ,f ) ◦ , u ( B,T,n • ,g ) ◦ ) − = ψ (cid:0) ( u ( A,S,m • ,f ) ◦ ) − , ( u ( B,T,n • ,g ) ◦ ) − (cid:1) = ( ρ A ∪ B,S ∪ TA,S , ρ A ∪ B,S ∪ TB,T ) − ◦ (cid:0) ( u ∗ × u ∗ ) ρ u ( A ∪ B ) ,u ( S ∪ T ) u ( A ) ,u ( S ) , ( u ∗ × u ∗ ) ρ u ( A ∪ B ) ,u ( S ∪ T ) u ( B ) ,u ( T ) (cid:1) = ( ρ A ∪ B,S ∪ TA,S , ρ A ∪ B,S ∪ TB,T ) − ◦ ( ρ A ∪ B,S ∪ TA,S , ρ A ∪ B,S ∪ TB,T ) ◦ ( u ∗ × u ∗ )= u ∗ × u ∗ = ( u ψ (( A,S,m • ,f ) , ( B,T,n • ,g )) ◦ ) − where we used (5 . ψ .Finally, E M -equivariance of ι amounts to saying that ι ( ∗ ) = ( ∅ , ∅ , ∅ ,
0) hasempty support, which is immediate from Lemma 3.7. (cid:3)
Proposition 5.5.
The functors ι and ψ define a lax symmetric monoidal structureon C • : E M -SSet τ → E M -Cat τ .Proof. It is trivial to check that ψ is natural; it remains to show the compatibilityof ψ and ι with the unitality, symmetry, and associativity isomorphisms. Unitality.
We will only prove left unitality, the argument for right unitality beinganalogous (in fact, right unitality will also follow from left unitality together with symmetry). For this we have to show that the composition ∗ ⊠ C X ι ⊠ C X −−−−→ C ∗ ⊠ C X ψ −→ C ∗ ⊠ X C λ −−→ C X agrees with the left unitality isomorphism of ( E M -Cat τ , ⊠ ), i.e. projection to thesecond factor.Let us first check this on objects: if ( A, S, m • , f ) ∈ C X is arbitrary, then theabove sends ( ∗ , ( A, S, m • , f )) by definition to ( ∅ ∪ A, ∅ ∪ A, ( ∅ ∪ m ) • , λ ◦ (0 ∪ f )),so the only non-trivial statement is that λ ◦ (0 ∪ f ) = f . Indeed, by definition0 ∪ f = (0 ◦ ρ A,S ∅ , ∅ , f ◦ ρ A,SA,S ). As λ : ∗ ⊠ X → X is given by projection to the secondfactor, we conclude λ ◦ (0 ∪ f ) = f ◦ ρ A,SA,S , so the claim follows from Lemma 5.2-(3).Next, let α : ( A, S, m • , f ) → ( B, T, n • , g ); we have to show that the above com-posite sends (id ∗ , α ) to α . As we already know that this has the correct source andtarget, it suffices to show this as morphism in E M -SSet . But indeed, plugging inthe definition we see that α is sent to(5.2) ( ρ A,S ∅ , ∅ , ρ A,SA,S ) − (id ◦ ρ A,S ∅ , ∅ , α ◦ ρ A,SA,S ) . As ρ A,SA,S = id by Lemma 5.2-(3), we see that projecting onto the second factor isleft inverse to ( ρ A,S ∅ , ∅ , ρ A,SA,S ); as the latter is an isomorphism by Lemma 5.2-(4) (oralternatively using that the projection is an isomorphism for obvious reasons), it isthen also right inverse, and (5 .
2) equals αρ A,SA,S = α as desired. Associativity.
We have to show that the diagram( C X ⊠ C Y ) ⊠ C Z C X ⊠ ( C Y ⊠ C Z ) C X ⊠ Y ⊠ C Z C X ⊠ C Y ⊠ Z C ( X ⊠ Y ) ⊠ Z C X ⊠ ( Y ⊠ Z ) a ∼ = ψ ⊠ C Z C X ⊠ ψψ ψ ∼ = C a commutes for all X, Y, Z ∈ E M -SSet τ ; here we denote the associativity isomor-phism by ‘ a ’ instead of the usual ‘ α ’ in order to avoid confusion with our notationfor a generic morphism in C • .To check this on objects we let (cid:0) (( A, S, m • , f ) , ( B, T, n • , g )) , ( C, U, o • , h ) (cid:1) be anyobject of the top left corner. Then the upper right path through the diagram sendsthis to ( A ∪ ( B ∪ C ) , S ∪ ( T ∪ U ) , ( m ∪ ( n ∪ o )) • , f ∪ ( g ∪ h )) while the lower left path sendsit to (( A ∪ B ) ∪ C, ( S ∪ T ) ∪ U, (( m ∪ n ) ∪ o ) • , a ◦ (( f ∪ g ) ∪ h ). It is clear that the firstthree components agree, so it only remains to show that f ∪ ( g ∪ h ) = a ◦ (( f ∪ g ) ∪ h )as maps E Inj( S ∪ T ∪ U, ω ) × Q i ∈ A ∪ B ∪ C ∆ ( m ∪ n ∪ o ) i → X ⊠ ( Y ⊠ Z ). But indeed, f ∪ ( g ∪ h ) = ( f ρ A ∪ B ∪ C,S ∪ T ∪ UA,S , ( g ∪ h ) ρ A ∪ B ∪ C,S ∪ T ∪ UB ∪ C,T ∪ U )= ( f ρ A ∪ B ∪ C,S ∪ T ∪ UA,S , ( gρ B ∪ C,T ∪ UB,T , hρ B ∪ C,T ∪ UC,U ) ρ A ∪ B ∪ C,S ∪ T ∪ UB ∪ C,T ∪ U )= ( f ρ A ∪ B ∪ C,S ∪ T ∪ UA,S , ( gρ A ∪ B ∪ C,S ∪ T ∪ UB,T , hρ A ∪ B ∪ C,S ∪ T ∪ UC,U ))where the final equality follows from Lemma 5.2-(2). Analogously, one shows that a ◦ (cid:0) ( f ∪ g ) ∪ h (cid:1) = a ◦ (( f ρ A ∪ B ∪ C,S ∪ T ∪ UA,S , gρ A ∪ B ∪ C,S ∪ T ∪ UB,T ) , hρ A ∪ B ∪ C,S ∪ T ∪ UC,U )and this is obviously equal to the above.
LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 35
Next, we let (cid:0) (( A ′ , S ′ , m ′• , f ′ ) , ( B ′ , T ′ , n ′• , g ′ )) , ( C ′ , U ′ , o ′• , h ′ ) (cid:1) be another suchobject, and we let (( α, β ) , γ ) be a morphism. We have to show that both pathsthrough the diagram send this to the same morphism in C X ⊠ ( Y ⊠ Z ) , for which itis then enough to show equality as morphisms in E M -SSet τ . For this we firstobserve that on the one hand by Lemma 5.2-(2)( ρ A ′ ∪ B ′ ∪ C ′ ,S ′ ∪ T ′ ∪ U ′ A ′ ,S ′ , ( ρ A ′ ∪ B ′ ∪ C ′ ,S ′ ∪ T ′ ∪ U ′ B ′ ,T ′ , ρ A ′ ∪ B ′ ∪ C ′ ,S ′ ∪ T ′ ∪ U ′ C ′ ,U ′ ))= (cid:0) id ⊠ ( ρ B ′ ∪ C ′ ,T ′ ∪ U ′ B ′ ,T ′ , ρ B ′ ∪ C ′ ,T ′ ∪ U ′ C ′ ,U ′ ) (cid:1) ◦ ( ρ A ′ ∪ B ′ ∪ C ′ ,S ′ ∪ T ′ ∪ U ′ A ′ ,S ′ , ρ A ′ ∪ B ′ ∪ C ′ ,S ′ ∪ T ′ ∪ U ′ B ′ ∪ C ′ ,T ′ ∪ U ′ )(in particular this is an isomorphism), and on the other hand obviously( ρ A ′ ∪ B ′ ∪ C ′ ,S ′ ∪ T ′ ∪ U ′ A ′ ,S ′ , ( ρ A ′ ∪ B ′ ∪ C ′ ,S ′ ∪ T ′ ∪ U ′ B ′ ,T ′ , ρ A ′ ∪ B ′ ∪ C ′ ,S ′ ∪ T ′ ∪ U ′ C ′ ,U ′ ))= a ◦ (cid:0) ( ρ A ′ ∪ B ′ ∪ C ′ ,S ′ ∪ T ′ ∪ U ′ A ′ ,S ′ , ρ A ′ ∪ B ′ ∪ C ′ ,S ′ ∪ T ′ ∪ U ′ B ′ ,T ′ ) , ρ A ′ ∪ B ′ ∪ C ′ ,S ′ ∪ T ′ ∪ U ′ C ′ ,U ′ (cid:1) . We now calculate(5.3) ( ρ A ′ ,S ′ , ( ρ B ′ ,T ′ , ρ C ′ ,U ′ )) ψ ( α, ψ ( β, γ ))= (id ⊠ ( ρ B ′ ,T ′ , ρ C ′ ,U ′ ))( αρ A,S , ψ ( β, γ ) ρ B ∪ C,T ∪ U )= ( αρ A,S , ( ρ B ′ ,T ′ , ρ C ′ ,U ′ ) ψ ( β, γ ) ρ B ∪ C,T ∪ U )= ( αρ A,S , ( βρ B,T , γρ
C,U ))where we omitted the superscripts for legibility. Analogously,( ρ A ′ ,S ′ , ( ρ B ′ ,T ′ , ρ C ′ ,U ′ )) ψ ( ψ ( α, β ) , γ ) = a ◦ (( ρ A ′ ,S ′ , ρ B ′ ,T ′ ) , ρ C ′ ,U ′ ) ψ ( ψ ( α, β ) , γ )= a ◦ (( αρ A,S , βρ
B,T ) , γρ C,U )which equals (5 . ψ ( α, ψ ( β, γ )) = ψ ( ψ ( α, β ) , γ ) as morphisms in E M -SSet as they agree after postcomposing with an isomorphism. This completesthe proof of associtativity. Symmetry.
Finally, we have to show that the diagram C X ⊠ C Y C Y ⊠ C X C X ⊠ Y C Y ⊠ Xτ ∼ = ψ ψ ∼ = C τ commutes for all tame E M -simplicial sets X, Y , where τ denotes the symmetryisomorphism of ⊠ on E M -Cat τ and E M -SSet τ , respectively; in both cases it isgiven by restriction of the flip map K × L ∼ = L × K .Again, let us first check this on objects. If (cid:0) ( A, S, m • , f ) , ( B, T, n • , g ) (cid:1) is anobject of the top left corner, then the upper right path through this diagram sendsthis to ( B ∪ A, T ∪ S, ( n ∪ m ) • , g ∪ f ), while the lower left path sends it to ( A ∪ B, S ∪ T, ( m ∪ n ) • , τ ◦ ( f ∪ g )). The first three components agree trivially, while forthe fourth components we simply calculate τ ◦ ( f ∪ g ) = τ ◦ ( f ρ A,S , gρ
B,T ) = ( gρ B,T , f ρ
A,S ) = g ∪ f. This proves commutativity on objects. If now (cid:0) ( A ′ , S ′ , m ′• , f ′ ) , ( B ′ , T ′ , n ′• , g ′ ) (cid:1) isanother such object and ( α, β ) is a morphism, then in order to show that bothpaths through the diagram send ( α, β ) to the same morphism of C Y ⊠ X it is againenough to check this as morphisms in E M -SSet . But indeed, the top right path through the diagram sends ( α, β ) to ( ρ B ′ ,T ′ , ρ A ′ ,S ′ ) − ( βρ B,T , αρ
A,S ). Using that( ρ B ′ ,T ′ , ρ A ′ ,S ′ ) = τ ◦ ( ρ A ′ ,S ′ , ρ B ′ ,T ′ ), this equals( ρ A ′ ,S ′ , ρ B ′ ,T ′ ) − ◦ τ ◦ ( βρ B,T , αρ
A,S ) = ( ρ A ′ ,S ′ , ρ B ′ ,T ′ ) − ( αρ A,S , βρ
B,T )which is by definition the image of ( α, β ) under the lower left composition. Thiscompletes the proof of symmetry and hence of the proposition. (cid:3)
As before, the corresponding result for C • : E M - G -SSet τ → E M - G -Cat τ follows formally. In particular, C • canonically lifts to a functor G -ParSumSSet → G -ParSumSSet . Explicitly, if X is a parsummable simplicial set, then C X has thesame underlying E M - G -category as before. The sum of two disjointly supportedobjects ( A, S, m • , f ) , ( B, T, n • , g ) is ( A ∪ B, S ∪ T, ( m ∪ n ) • , f + g ), with f + g givenby the composition E Inj( S ∪ T, ω ) × Y i ∈ A ∪ B ∆ ( m ∪ n ) i f ∪ g −−→ X ⊠ X + −→ X where + denotes the sum operation of the parsummable G -simplicial set X . More-over, the sum of two morphisms α , β having disjointly supported sources anddisjointly supported targets agrees as a map of E M -simplicial sets with ψ ( α, β ) asdefined above. Finally, the unit is given by ( ∅ , ∅ , ∅ ,
0) where 0 denotes the map E Inj( ∅ , ω ) × Q ∅ → X with image the zero vertex of X .Next, we will show that the natural maps ǫ and ˜ ǫ also define natural transfor-mations between these lifts. Proposition 5.6.
The natural transformation ǫ : N ◦ C • ⇒ id E M -SSet τ is (sym-metric) monoidal.Proof. We have to show that the diagrams(N C X ) ⊠ (N C Y ) X ⊠ Y N C X ⊠ Y ǫ ⊠ ǫψ ǫ ∗ ∗ N C ∗ = ι ǫ commute, where ψ and ι denote the compositions of the structure maps of N and C • of the same name.The commutativity of the right hand triangle is trivial as the target is terminal.For the left hand triangle, we consider any k -simplex of (N C X ) ⊠ (N C Y ). This isby definition and Example 2.7 given by a pair of a k -simplex( A , S , m (0) • , f ) α −→ · · · α k −−→ ( A k , S k , m ( k ) • , f k )of N( C X ) and a k -simplex( B , T , n (0) • , g ) β −→ · · · β k −→ ( B k , T k , n ( k ) • , g k )of N( C Y ) such that supp( A i , S i , m ( i ) • , f i ) ∩ supp( B i , T i , n ( i ) • , g i ) = ∅ for i = 0 , . . . , k .If σ α • , σ β • are defined as before, then the top arrow in this diagram sends( α • , β • ) to ( f k ( σ α • ) , g k ( σ β • )). On the other hand, the lower path sends ( α • , β • ) to( f k ∪ g k )( σ ψ ( α • ,β • ) ). Here σ ψ ( α • ,β • ) is uniquely characterized by demanding thatits ℓ -th vertex be given by ψ ( α k , β k ) · · · ψ ( α ℓ +1 , β ℓ +1 )( ι S ℓ ∪ T ℓ , ∗ ) ∈ E Inj( S k ∪ T k , ω ) × Y i ∈ A k ∪ B k ∆ ( m ∪ n ) i . LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 37
By functoriality of ψ and its definition, this is equal to( ρ A k ,S k , ρ B k ,T k ) − (cid:0) ( α k · · · α ℓ +1 ) ⊠ ( β k · · · β ℓ +1 ) (cid:1) ( ρ A ℓ ,S ℓ , ρ B ℓ ,T ℓ )( ι S ℓ ∪ T ℓ , ∗ ) , and as obviously ( ρ A ℓ ,S ℓ , ρ B ℓ ,T ℓ )( ι S ℓ ∪ T ℓ , ∗ ) = (cid:0) ( ι S ℓ , ∗ ) , ( ι T ℓ , ∗ ) (cid:1) , we conclude that σ ψ ( α • ,β • ) = ( ρ A k ,S k , ρ B k ,T k ) − ( σ α • , σ β • ). Thus, ǫ ( ψ ( α • , β • )) = ( f k ∪ g k )( σ ψ ( α • ,β • ) ) = ( f k ⊠ g k )( σ α • , σ β • ) = ( f k ( σ α • ) , g k ( σ β • ))as claimed. (cid:3) Proposition 5.7.
The natural transformation ˜ ǫ : C • ◦ N ⇒ id E M -Cat τ is (sym-metric) monoidal.Proof. We have to prove commutativity of the diagrams C N C ⊠ C N D C ⊠ D C N( C ⊠ D ) ˜ ǫ ⊠ ˜ ǫψ ˜ ǫ ∗ ∗ C N( ∗ ) = ι ˜ ǫ which in the case of the right hand triangle is trivial again. For the left handdiagram, it suffices to prove this after applying N as the latter is fully faithful. Theresulting diagram is(5.4) N( C N C ⊠ C N D ) N( C ⊠ D )N( C N( C ⊠ D ) ) N(˜ ǫ ⊠ ˜ ǫ )N( ψ ) ǫ where we have applied the definition of ˜ ǫ . We now consider the three-dimensionaldiagram N( C N C ⊠ C N D ) N( C ⊠ D )(N C N C ) ⊠ (N C N D ) (N C ) ⊠ (N D )N C N( C ⊠ D ) N C (N C ) ⊠ (N D ) ∼ = ψ ∼ = ψ N C ψ ∼ = where the back face is (5 . ǫ : N ◦ C • ⇒ id, and the front-to-back maps are induced by the structure isomorphisms of thestrong symmetric monoidal functor N as indicated. Then the front face commutesby the previous proposition, the left face commutes by the definition of the structuremaps of a composition of lax symmetric monoidal functors, the top face commutesby naturality of ψ , and the lower right face commutes by naturality of ǫ . As all thefront-to-back maps are isomorphisms, it follows that also the back face commutes,which then completes the proof of the proposition. (cid:3) As before, we automatically get the corresponding statements for the lifts of ǫ and ˜ ǫ to E M - G -SSet τ and E M - G -Cat τ , respectively. We can now immediatelyprove the following precise form of Theorem B from the introduction: Theorem 5.8.
The lifts of N and C • constructed above define mutually inversehomotopy equivalences C • : G -ParSumSSet ⇄ G -ParSumCat : N with respect to the G -global weak equivalences on both sides. More precisely, thenatural maps ǫ : N( C X ) → X and ˜ ǫ : C N C → C define levelwise G -global weakequivalences between the two composites and the respective identities.Proof. We first observe that these indeed assemble into natural transformationsN ◦ C • ⇒ id G -ParSumSSet and C • ◦ N ⇒ id G -ParSumCat by Propositions 5.6 and 5.7,respectively. As the weak equivalences of G -ParSumSSet and G -ParSumCat arecreated in E M - G -SSet τ and E M - G -Cat τ , respectively, the claim now followsfrom Theorem 4.1. (cid:3) Moreover, we can now prove:
Theorem 5.9.
All the inclusions in G -ParSumCat s ֒ → G -ParSumCat ws ֒ → G -ParSumCat are homotopy equivalences with respect to the G -global weak equivalences.Proof. For the left hand inclusion we have already shown this as Corollary 1.33. Forthe right hand inclusion it suffices to observe again that C • ◦ N is homotopy inverse,which follows from Theorem 5.8 by the same arguments as in Corollary 4.7. (cid:3) G -Global Homotopy Theory of G -Symmetric Monoidal Categories We write
SymMonCat for the 1-category of small symmetric monoidal cate-gories and strong symmetric monoidal functors, and we denote by G -SymMonCat the corresponding category of G -objects. Explicitly, an object of G -SymMonCat is a symmetric monoidal category equipped with a strict G -action through strongsymmetric monoidal functors, and the morphisms are given by strong symmetricmonoidal functors that strictly preserve the actions.We want to study G -SymMonCat from a G -global perspective, for which weintroduce the following notion of weak equivalence: Definition 6.1. A G -equivariant functor f : C → D of small G -categories is calleda G -global weak equivalence if Fun H ( EH, f ) : Fun H ( EH, ϕ ∗ C ) → Fun H ( EH, ϕ ∗ D )is a weak homotopy equivalence for every finite group H and every homomorphism ϕ : H → G . A morphism in G -SymMonCat is called a G -global weak equivalenceif and only if its underlying G -equivariant functor is.By Lemma 1.24 we may restrict ourselves to those H that are universal sub-groups of M in the above definition without changing the notion of G -global weakequivalence. Example . If G = 1, then the 1-global weak equivalences of small categories areprecisely the global equivalences in the sense of [Sch19a, Definition 3.2]. Example . Any underlying equivalence of categories induces equivalences oncategorical homotopy fixed points, so it is in particular a G -global weak equivalence. LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 39
Remark . The G -global weak equivalences might look a bit counterintuitive atfirst, so let us explain the connection to classical equivariant K -theory, for whichwe assume that G is finite.If C is a small symmetric monoidal category, then the Shimada-Shimakawa con-struction [SS79, Definition 2.1] associates to this a special Γ-space S ( C ). If wenow let G act suitably on C , then S ( C ) acquires a G -action through functoriality,making it into a Γ- G -space.Unfortunately, S ( C ) is usually not special in the correct G -equivariant sense.However, it is an observation going back to Shimakawa [Shi89, discussion beforeTheorem A ′ ] and later extensively used by [Mer17] that this defect can be cured byreplacing C with Fun( EG, C ) equipped with the conjugation action.Thus, the natural way to obtain a special Γ- G -space is via the composition(6.1) G -SymMonCat Fun(
EG, –) −−−−−−−→ G -SymMonCat S −→ Γ- G -SSet , and this is also the basis for the usual definition of the equivariant algebraic K -theory of C .There are several useful notions of G -equivariant weak equivalences on the righthand side, the simplest (and strongest) of which are the level equivalences , seee.g. [Ost16, 4.2.1]. It is then not hard to check from the definitions that a G -equivariant strong symmetric monoidal functor f : C → C ′ induces a level equiva-lence under (6 .
1) if and only if the induced functor Fun(
EG, f ) H : Fun( EG, C ) H → Fun(
EG, C ′ ) H is a weak homotopy equivalence for every subgroup H ⊂ G . Usingthat the inclusion H ֒ → G induces an H -equivariant equivalence EH → EG , weconclude in particular that any G -global weak equivalence induces a level equiva-lence under (6 . G -global homotopy theory of G -SymMonCat to the models considered so far, for which it will be useful to introduce an interme-diate step. We therefore recall: Definition 6.5. A permutative category is a symmetric monoidal category in whichthe associativity and unitality isomorphisms are the respective identities. We write PermCat for the category of small permutative categories and strict symmetricmonoidal functors.It is well-known that the inclusion
PermCat ֒ → SymMonCat is a homo-topy equivalence with respect to the underlying equivalences of categories (also see[Len20, Theorem 1.18] for a sketch why this is a consequence of Mac Lane’s Coher-ence Theorem). Thus, roughly speaking,
PermCat is just as good as
SymMonCat from a purely formal point of view. In practice, however, working with permutativecategories is often easier than working with general symmetric monoidal categoriesas there are less coherence data to keep track of.As a concrete manifestation of this, Schwede constructs in [Sch19b, Construc-tion 11.1] an explicit functor Φ :
PermCat → ParSumCat , and while it is plau-sible that his construction could be extended to all small symmetric monoidalcategories, working out the details would probably become quite technical andcumbersome. Accordingly, the parsummable categories associated to general smallsymmetric monoidal categories are only defined indirectly by applying Φ to a per-mutative replacement.
If we look at parsummable categories from a categorical angle, then Φ is verywell-behaved. Namely, we proved as the main result of [Len20]:
Theorem 6.6.
The functor
Φ :
PermCat → ParSumCat is a homotopy equiva-lence with respect to the underlying equivalences of categories on both sides.Proof.
This is [Len20, Theorem 3.26]. (cid:3)
However, from a global perspective Φ( C ) is not yet the ‘correct’ parsummablecategory associated to C . For example, [Sch19b, Proposition 11.9] implies that if C is any small permutative replacement of the symmetric monoidal category of finitedimensional C -vector spaces and C -linear isomorphisms under ⊕ , then the global K -theory of Φ( C ) is different from the usual definition of the global algebraic K -theory K gl ( C ) of the complex numbers. In order to avoid this issue, one appliesthe saturation construction first, so that the global K -theory of C is obtained byfeeding Φ( C ) sat into Schwede’s machinery.Thus, if we write G -PermCat for the category of G -objects in PermCat , thenit is actually the composition(6.2) G -PermCat Φ −→ G -ParSumCat (–) sat −−−→ G -ParSumCat that is the natural way to associate a G -parsummable category to a small G -permutative category, at least from a G -global point of view. We therefore want toprove: Theorem 6.7.
The composition (6 . is a homotopy equivalence with respect tothe G -global weak equivalences on both sides.Remark . As the inclusion defines a homotopy equivalence between
PermCat and
SymMonCat with respect to the underlying equivalences of categories, itfollows formally that it also induces a homotopy equivalence G -PermCat ֒ → G -SymMonCat with respect to the underlying equivalences. Since it moreoverpreserves and reflects G -global weak equivalences by definition, Example 6.3 im-plies that this remains a homotopy equivalence (with the same homotopy inverseas before) when we equip both sides with the G -global weak equivalences insteadof the underlying equivalences of categories. The above theorem is therefore theappropriate G -global generalization of Theorem A from the introduction. We canmoreover conclude from it that if b Φ : G -SymMonCat → G -ParSumCat is any(hypothetical) extension of Φ respecting underlying equivalences of categories, then(–) sat ◦ b Φ : G -SymMonCat → G -ParSumCat is a homotopy equivalence with re-spect to the G -global weak equivalences on both sides. Remark . Elaborating on Schwede’s argument mentioned above, we showedin [Len20, Remark 3.27] that there is no small permutative category C at all suchthat the global algebraic K -theory of Φ( C ) is equivalent to K gl ( C ), which in par-ticular implies that there is no notion of weak equivalence on PermCat such thatΦ becomes an equivalence of homotopy theories with respect to the global weakequivalences on
ParSumCat , and this even remains impossible when we pass tothe larger class of those morphisms that induce global weak equivalences on K -theory. Thus, the passage to saturations is not a mere artifact of our proof (or ourprejudices against the parsummable categories Φ( C ) and their global K -theory),but actually necessary. LOBAL HOMOTOPY THEORY OF SYMMETRIC MONOIDAL CATEGORIES 41
For the proof of the theorem we need the following lemma, which allows us toget rid of the additional E M -action in the definition of the ‘homotopy’ fixed pointsof G -parsummable categories: Lemma 6.10.
Let f : C → D be a morphism in E M - G -Cat , let H ⊂ M be anysubgroup, and let ϕ : H → G be any homomorphism. Then f ‘ h ’ ϕ is a weak homotopyequivalence if and only if Fun H ( EH, ϕ ∗ C ) is.Proof. We will show that C ‘ h ’ ϕ and Fun H ( EH, ϕ ∗ C ) are connected by a naturalzig-zag of equivalences of categories, which will then immediately imply the lemma.While the claim could be proven analogously to [Sch19b, Proposition 7.6], weprefer a slightly different argument: let us consider the zig-zag(6.3) C action ←−−−− E M × C triv pr −→ C triv where C triv has the same underlying G -category, but trivial E M -action, and E M acts on itself from the left in the obvious way. There is an evident way to make themiddle term functorial in C , and with respect to this the above two maps are clearlynatural. Moreover, one easily checks that they are both ( E M × G )-equivariant.We now claim that they are also underlying equivalences of categories. Indeed,this is obvious for the projection as E M is contractible. The non-equivariantfunctor (1 , –) : C → E M × C is right-inverse to it, hence again an equivalence ofcategories. But it is also right inverse to the action map E M × C → C , hence alsothe latter is an equivalence of categories as desired.The claim now simply follows by applying (–) ‘ h ’ ϕ to (6 . (cid:3) Proof of Theorem 6.7.
With Theorem 5.9 in place one can now argue precisely aswe sketched in [Len20, Remark 3.27] (for G = 1):By Theorem 1.30, the functor (6 .
2) factors through the inclusion of the full sub-category G -ParSumCat s , and as the latter is a homotopy equivalence with respectto the G -global weak equivalences by Theorem 5.9, it is then enough to show that(6 .
2) is a homotopy equivalence when viewed as a functor into G -ParSumCat s .This is true with respect to the categorical equivalences by Theorem 6.6 togetherwith Corollary 1.32; moreover, the G -global weak equivalences on G -PermCat are finer than the categorical ones by Example 6.3, and so are the G -global weakequivalences on G -ParSumCat s by Lemma 1.28. To finish the proof, it is thereforeenough to show that (–) sat ◦ Φ preserves and reflects G -global weak equivalences.For this we fix a universal subgroup H ⊂ M together with a homomorphism ϕ : H → G . If now f : C → D is a G -equivariant strict symmetric monoidal func-tor, then Theorem 1.30 implies that Φ( f ) sat induces a weak homotopy equiva-lence on ϕ -fixed points if and only if Φ( C ) ‘ h ’ ϕ → Φ( D ) ‘ h ’ ϕ is a weak homotopyequivalence, which is in turn equivalent to Fun H ( EH, f ) : Fun H ( EH, ϕ ∗ Φ( C )) → Fun H ( EH, ϕ ∗ Φ( D )) being a weak homotopy equivalence by the previous lemma.Finally, by [Sch19b, Remark 11.4] we have natural equivalences of categories Φ( C ) ≃ C , Φ( D ) ≃ D , and these are automatically G -equivariant as the G -actions on theleft hand sides are induced by functoriality of Φ.Thus, we altogether see that Φ( f ) sat induces a weak homotopy equivalence on ϕ -fixed points if and only if Fun H ( EH, f ) : Fun H ( EH, ϕ ∗ C ) → Fun H ( EH, ϕ ∗ D ) isa weak homotopy equivalence. Letting ϕ vary, this precisely yields the definitions ofthe G -global weak equivalences on G -ParSumCat and G -PermCat , respectively,which completes the proof of the theorem. (cid:3) References [Cis19] Denis-Charles Cisinski.
Higher categories and homotopical algebra. , volume 180 of
Camb.Stud. Adv. Math.
Cambridge: Cambridge University Press, 2019.[Ill72] Luc Illusie.
Complexe cotangent et d´eformations. II. , volume 283 of
Lect. Notes Math.
Berlin-Heidelberg-New York: Springer-Verlag, 1972.[Len] Tobias Lenz. G -global homotopy theory and algebraic K -theory. In preparation.[Len20] Tobias Lenz. Parsummable categories as a strictification of symmetric monoidal cate-gories. Preprint, available as arXiv:2006.15068 , 2020.[Man10] Michael A. Mandell. An inverse K -theory functor. Doc. Math. , 15:765–791, 2010.[Mer17] Mona Merling. Equivariant algebraic K -theory of G -rings. Math. Z. , 285(3-4):1205–1248,2017.[MM19] Cary Malkiewich and Mona Merling. Equivariant A -theory. Doc. Math. , 24:815–855,2019.[Ost16] Dominik Ostermayr. Equivariant Γ-spaces.
Homology Homotopy Appl. , 18(1):295–324,2016.[Qui71] Daniel Quillen. Cohomology of groups. In
Actes Congr. internat. Math. 1970 , volume 2,pages 47–51, 1971.[Qui73] Daniel G. Quillen. Higher algebraic K -theory. I. In Algebr. K -Theory I, Proc. Conf.Battelle Inst. 1972 , volume 341 of Lect. Notes Math. , pages 85–147, 1973.[Sch08] Stefan Schwede. On the homotopy groups of symmetric spectra.
Geom. Topol. ,12(3):1313–1344, 2008.[Sch18] Stefan Schwede.
Global homotopy theory , volume 34 of
New Math. Monogr.
Cambridge:Cambridge University Press, 2018.[Sch19a] Stefan Schwede. Categories and orbispaces.
Algebr. Geom. Topol. , 19(6):3171–3215, 2019.[Sch19b] Stefan Schwede. Global algebraic K -theory. Preprint, available as arxiv:1912.08872 .,2019.[Sch20] Stefan Schwede. Orbispaces, orthogonal spaces, and the universal compact Lie group. Math. Z. , 294(1-2):71–107, 2020.[Seg74] Graeme Segal. Categories and cohomology theories.
Topology , 13:293–312, 1974.[Shi89] Kazuhisa Shimakawa. Infinite loop G -spaces associated to monoidal G -graded categories. Publ. Res. Inst. Math. Sci. , 25(2):239–262, 1989.[SS79] Nobuo Shimada and Kazuhisa Shimakawa. Delooping symmetric monoidal categories.
Hiroshima Math. J. , 9:627–645, 1979.[SS20] Steffen Sagave and Stefan Schwede. Homotopy invariance of convolution products. Toappear in
Int. Math. Res. Not. , published online doi:10.1093/imrn/rnz334 ., 2020.[Tho95] R. W. Thomason. Symmetric monoidal categories model all connective spectra.
TheoryAppl. Categ. , 1:78–118, 1995.
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