aa r X i v : . [ m a t h . A T ] J u l G ∞ -ring spectra and Moore spectra for β -rings Michael StahlhauerJuly 29, 2020
Abstract
In this paper, we introduce the notion of G ∞ -ring spectra. These are globally equivarianthomotopy types with a structured multiplication, giving rise to power operations on theirequivariant homotopy and cohomology groups. We illustrate this structure by analysingwhen a Moore spectrum can be endowed with a G ∞ -ring structure. Such G ∞ -structurescorrespond to power operations on the underlying ring, indexed by the Burnside ring. Weexhibit a close relation between these globally equivariant power operations and the structureof a β -ring, thus providing a new perspective on the theory of β -rings. Contents
Introduction 21 The symmetric algebra monad 52 Power operations on ultra-commutative ring spectra 9 G ∞ -ring spectra and their properties 20 G ∞ -ring spectra and their power operations . . . . . . . . . . . . . 203.1.1 External power operations via representability . . . . . . . . . . . . . . . 223.2 An adjunction between G ∞ - and H ∞ -ring spectra . . . . . . . . . . . . . . . . . 243.2.1 Lifting the forgetful functor GH → SH to structured ring spectra . . . . . 253.2.2 Lifting the right adjoint
SH → GH to structured ring spectra . . . . . . . 273.3 Homotopical analysis of the extended powers . . . . . . . . . . . . . . . . . . . . 32 β -rings 36 F ⊗ F ( e ) B . . . . . . . . . . . . . . . . . . . 364.2 The relation to β -rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3 G ∞ -structure on Moore spectra for powered algebras . . . . . . . . . . . . . . . . 48 A Transfer results for monads and comonads 56
A.1 Transferring monads under lax functors . . . . . . . . . . . . . . . . . . . . . . . 56A.2 Transferring a comonad along an adjunction . . . . . . . . . . . . . . . . . . . . . 63
References 68 for a list of codes.The research in this paper was supported by the DFG Schwerpunktprogramm 1786 HomotopyTheory and Algebraic Geometry (GZ SCHW 860/1-1) and the Max-Planck-Institut fürMathematik Bonn. The author is an Associate Member of the Hausdorff Center forMathematics at the University of Bonn (DFG GZ 2047/1, project ID 390685813). ntroduction
The aim of this article is the introduction of the new notion of G ∞ -ring spectra. These supportpower operations on their homotopy groups and cohomology with coefficients in such spectra.We moreover provide an algebraic description of this notion on Moore spectra, linking G ∞ -ringstructures on a Moore spectrum to β -ring structures on the represented ring.Algebraic invariants are more useful the more structure they are endowed with. One ex-ample of this slogan are power operations on cohomology. The Steenrod operations on mod- p cohomology, the Adams-operations on K -theory and power operations on complex bordism andcohomotopy all carry a lot of additional information and have seen extensive use in classicalhomotopy theory. More recently, Hill, Hopkins and Ravenel used power operations in the guiseof norm maps to prove the non-existence of elements of Kervaire-invariant one in [19]. This workrenewed interest in both multiplicative aspects of homotopy theory and equivariant techniques.Classical power operations in cohomology arise from an H ∞ -ring structure on the representingspectrum, as defined by Bruner, May, McClure and Steinberger in [12]. In the present work,we generalize the notion of an H ∞ -ring spectrum to a globally equivariant context, in orderto represent equivariant power operations. Here, globally equivariant means that we encodecompatible actions by all compact Lie groups, using the framework provided by Schwede in [34].Thus, a G ∞ -ring spectrum encodes power operations on equivariant cohomology groups for allcompact Lie groups. Hence, the notion of a G ∞ -ring spectrum relates to the stricter notion ofan ultra-commutative ring spectrum as an H ∞ -ring structure relates to an E ∞ -ring spectrum.Algebraically, power operations can be packaged in different ways. The Adams-operations on K -theory endow it with the structure of a λ -ring, and the power operations on cohomotopy giveit the structure of a β -ring. Among these, λ -rings are better-behaved and are widely studied inalgebraic topology and representation theory, e.g. [20, 5, 23]. On the other hand, the theory of β -rings is still largely mysterious, with different definitions and many subtleties not present in thestudy of λ -rings, see e.g. [33, 31, 43]. In this paper, we present a different approach to the notionof β -rings, coming from a well-structured theory of global power operations, where the questionof scalar extensions of the Burnside ring global power functor naturally leads to considering β -rings. In this way, G ∞ -ring structures on Moore spectra, which yield scalar extensions of theglobal power operations on the sphere spectrum, are intimately tied to β -ring structures. Results
In the first part of this work, we introduce the notion of a G ∞ -ring spectrum. Thisis a derived version of a structured ring spectrum. In contrast to e.g. an E ∞ -ring spectrum, thedefinition is at the level of the homotopy category. Concretely, we take the free commutativealgebra monad P : S p → S p at the level of spectra. This functor is left derivable for the positiveglobal model structure and thus induces a monad on the global homotopy category GH . Definition (Definition 3.3) . A G ∞ -ring spectrum is an algebra over the monad G = L P .We then study properties of G ∞ -ring spectra. As mentioned above, the main property of a G ∞ -ring spectrum is that it supports power operations on its equivariant homotopy groups: Theorem (Proposition 3.8) . Let E be a G ∞ -ring spectrum. The structure map G E → E definesthe structure of a global power functor on π ( E ) . To prove this proposition, we construct external power operations π G ( E ) → π Σ m ≀ G ( G E ),which exist for any global spectrum E . In presence of a G ∞ -structure, the power operations onthe homotopy groups are then obtained by postcomposition with the multiplication.2ny ultra-commutative ring spectrum induces a G ∞ -ring structure on its homotopy type. How-ever, there are also examples of G ∞ -ring spectra which are not induced from any strictly com-mutative multiplication. These are constructed by means of an adjunction G ∞ -Rings H ∞ -Rings , UR where U is the forgetful functor from G ∞ -rings to H ∞ -rings and R does not change the underlying H ∞ -ring spectrum. Applying the right adjoint R to an H ∞ -ring spectrum E which cannot berigidified to a strict commutative ring spectrum, we see that RE is not the homotopy type of anultra-commutative ring spectrum.In the second part of this paper, we analyse the structure of a G ∞ -multiplication on Moorespectra. For this, we first consider scalar extensions of a global power functor R . This leadsto the notion of an R -powered algebra, which describes a ring together with power operationsgoverned by the equivariant structure of R . We obtain the following result, which characterizesthe G ∞ -ring structures on global Moore spectra via purely algebraic data: Theorem (Theorem 4.43) . The functor π e : G ∞ -Moore torsion-free → P owAlg torsion-free A is an equivalence of categories between the homotopy category of G ∞ -Moore spectra for countabletorsion-free rings and the category of countable torsion-free A -powered algebras. Here, A denotes the Burnside ring global power functor. The restriction to torsion-free ringsis necessary, since already the existence of multiplications on Moore spectra in the presenceof torsion is a subtle question. Countability is a mild technical assumption we use in orderto construct such Moore spectra. The above theorem shows that whenever multiplications onMoore-spectra are tractable, then also the power operations are completely determined by alge-braic power operations on the represented ring. Moreover, the theory of A -powered algebras isclosely linked to the theory of β -rings. Theorem (Theorem 4.30) . The assignment ( G, C ) A ( G ) ⊗ C extends to a functor A ( _ ) ⊗ _ : Rep op × P owAlg A → β -Rings , which sends a conjugacy class of a morphism of compact Lie groups to the corresponding restric-tion. Here, Rep is the category of compact Lie groups and conjugacy classes of continuous homo-morphisms between these. In particular, this shows that for a Moore spectrum S B that supportsa G ∞ -ring structure, all equivariant homotopy groups π G ( S B ) come endowed with the structureof a β -ring.The above theorem shows that the notion of an A -powered algebra captures the structure ofa ring supporting β -ring structures at all compact Lie groups at once. Structure
In Section 1, we fix notation for the symmetric algebra monad, which we use todefine G ∞ -ring spectra.In Section 2, we recall the basics on orthogonal spectra as models for global homotopy theory,and the multiplicative aspects leading to power operations. Moreover, we give a construction ofthe external power operations on the equivariant homotopy groups of any spectrum.In Section 3, we define G ∞ -ring spectra and see that they support power operations on their3omotopy groups. We also compare this new notion to classical H ∞ -ring spectra by means of anadjunction featuring the forgetful functor, and provide a homotopical comparison of the derivedsymmetric algebra monad G = L P and extended symmetric powers Σ ∞ + ( E gl Σ m ) ∧ Σ m X ∧ m .In the last section, we analyse G ∞ -ring Moore spectra. On the algebraic side, we consider scalarextensions of global power functors, giving rise to the definition of A -powered algebras, andexhibit their close relation to β -rings. On the topological side, we prove that an A -poweredalgebra structure on B is equivalent to a G ∞ -ring structure on S B for torsion-free B .In the appendix, we collect results from the theory of monads and comonads used throughoutour work. In particular, we study under which 2- and double categorical functors monads arepreserved. We utilize these results when constructing the adjunction between G ∞ - and H ∞ -ringspectra. Moreover, we consider a result transferring comonads under adjunctions. Acknowledgements
This paper is a revised version of a Master’s thesis written at the Uni-versity of Bonn. I would like to thank my supervisor Stefan Schwede for suggesting the topic of G ∞ -ring spectra and β -rings to me and for his constant support and encouragement.I would also like to thank Markus Hausmann for helpful remarks, Irakli Patchkoria for answer-ing questions on Moore spectra and Jack Davies for interesting conversations on global powerfunctors. 4 The symmetric algebra monad
In this paper, we freely use the concepts of monads and algebras over them. For an introduction,we refer to [27, Chapter VI]. Monads are used as an organizational tool in order to compactlypackage the algebraic structure we consider, namely that of a structured multiplication on aglobal homotopy type. The main example of a monad in this work is the symmetric algebramonad, defining commutative monoids in a symmetric monoidal category, and we fix notation inthis section. We discuss what structure is needed in order to obtain an algebra over this monadin Proposition 1.3, and which functors induce functors between the algebras over a symmetricalgebra monad in Lemma 1.7. This is accomplished by introducing the abstract notion of amonad functor between categories endowed with a monad.
Definition 1.1.
Let ( C , ⊗ ,
1) be a cocomplete closed symmetric monoidal category. We set forany object X in C and m ≥ m -th symmetric power as P m X = X ⊗ m / Σ m , and P X = 1 asthe unit of ⊗ . The total symmetric power functor is P X = _ m ≥ P m X, where we use the symbol ∨ for the coproduct in C .For a morphism f : X → Y , we define P f : P X → P Y as P f = W m ≥ f ⊗ m / Σ m .We can define a multiplication map P X ⊗ P X → P X , using the concatenation map X ⊗ m ⊗ X ⊗ n ∼ = X ⊗ m + n after passage to the orbits. This multiplication map together with the inclusionof the unit as P X makes P X into a commutative monoid in the category C . Moreover, forming P f : P X → P Y for a map f : X → Y gives a morphism of monoids. Thus the functor P : C → C factors through the category Com of commutative monoids in C . We also call this functor P : C →
Com by abuse of notation.Note that this functor is left adjoint to the forgetful functor Com → C , with unit X → P X, X = P X → P X given as the inclusion of the linear factor and counit the multiplication map P Y → Y for a commutative monoid Y .As we have an adjunction, this defines a monad P = U ◦ P : C → C in the category C . We canthen form the category of P -algebras. Lemma 1.2.
The category
Com of commutative monoids and the category of P -algebras areisomorphic, via the functor which defines for a commutative monoid X a P -algebra structure asthe multiplication map P X → X of the above adjunction, and which is the identity on morphisms. Thus we can use the symmetric algebra monad to give an alternative definition of commutativemonoids, which incorporates not only the commutative multiplication itself but also the coherenceof the higher multiplications. This reformulation is useful to us, as we later pass to the homotopycategory of a symmetric monoidal model category. If we can derive the symmetric power monad5n this context, the two notions of commutative monoids and algebras over the derived monadusually do not coincide. This phenomenon appears in our context.In specifying an algebra structure over P , it is often convenient to consider the levels P m ofthe monad separately and to decompose the required map h = W m ≥ h m . Thus, we need totranslate the properties of being an algebra over P into this context. This uses the following twomaps:For i, j ≥
0, we define ϕ i,j : P i X ⊗ P j X ∼ = X ⊗ i + j / (Σ i × Σ j ) → X ⊗ i + j / Σ i + j = P i + j X as the projection along the inclusion Σ i × Σ j → Σ i + j , and for k, m ≥
0, we define ψ k,m : P k P m X ∼ = X ⊗ km / (Σ k ≀ Σ m ) → X ⊗ km / Σ km = P km X as the projection along the inclusion of block permutations. Here, the wreath product Σ k ≀ Σ m is the semi-direct product Σ k ⋉ Σ km with respect to the permutation action of Σ k on Σ km . Proposition 1.3.
Let X be an object of C , and h = W m ≥ h m : P X → X be a morphism. Thenthe following are equivalent:a) The map h makes X into a P -algebra.b) The maps h m : P m X → X satisfy the following relations:i) h = Id under the identification P X ∼ = X .ii) For all i, j ≥ , the diagram P i X ⊗ P j X X ⊗ X P X P i + j X X h i ⊗ h j ϕ i,j pr h h i + j commutes.iii) For all k, m ≥ , the diagram P k P m X P k X P km X X P k h m ψ k,m h k h km commutes.Proof. For the translation, we need to make the associativity transformation µ : PP → P explicit.To this end, we calculate PP X as follows: PP X = _ m ≥ P m ( P X ) = _ m ≥ P m _ k ≥ P k X = _ m ≥ _ k ≥ P k X ⊗ m / Σ m = _ m ≥ _ sequences ( ij )with P ij = m ( P k X ) ⊗ i k / Σ i k ⊗ . . . ⊗ ( P k n X ) ⊗ i kn / Σ i kn ∼ = _ sequences ( ij )with P ij finite P i k P k X ⊗ . . . ⊗ P i kn P k n X Here, the numbers k j in the last two lines are exactly those such that i k j = 0. Using thisrepresentation, we see that the concatenation morphism µ : PP X → P X featuring in the monadstructure of P is given on each wedge-summand as the map P i k P k X ⊗ . . . ⊗ P i kn P k n X ⊗ ψ ikj ,kj −−−−−−→ P i k k X ⊗ . . . ⊗ P i kn k n X ϕ −→ P Σ i kj k j X incl −−→ P X. Moreover, the unit of the monad is η : X ∼ = P X → P X , the inclusion as the linear summand.Using this explicit description of the monad structure, we see that the unit diagram for X isequivalent to the property h = Id, and the associativity diagram is equivalent to the other twodiagrams ii ) and iii ) above. This proves the proposition.Since we use monads to describe commutative monoids, we also investigate which functorslift to categories of algebras in this context. To this end, the concept of a lax monad functor isintroduced in [38, §1]. We give a recollection of the definitions and results. Definition 1.4.
Let C and D be categories and ( P, µ, η ) and (
Q, ν, ε ) be monads in C and D respectively. A (lax) monad functor is a pair ( F, ρ ), consisting of a functor F : C → D and anatural transformation ρ : QF → F P , such that the diagrams
F QFF P εFF η ρ and
QQF QF PF P PQF F P
QρνF ρPF µρ commute.Let (
F, ρ ) , ( G, σ ) :
C → D be two monad functors between P and Q . Then a monadic naturaltransformation between F and G is a natural transformation θ : F → G such that θP ◦ ρ = σ ◦ Qθ as natural transformations QF → GP . Proposition 1.5.
Let C and D be categories and ( P, µ, η ) and ( Q, ν, ε ) be monads on C and D respectively. Let ( F, ρ ) :
C → D be a lax monad functor. Then the assignment ( X, h : P X → X ) ( F X, F ( h ) ◦ ρ : QF X → F P X → F X ) extends to a functor F : Alg P → Alg Q . Moreover, let ( F, ρ ) and ( G, σ ) be monad functors between P and Q and let θ be a monadicnatural transformation between F and G . Then θ lifts to a natural transformation between thefunctors F, G : Alg P → Alg Q . Lemma 1.6.
Let C and D be categories, P be a monad on C and Q be a monad on D . Let ( F, ρ ) , ( G, σ ) :
C → D be two monad functors between P and Q . Let θ : F → G be a monadictransformation which is invertible as a natural transformation. Then also the inverse θ − : G → F is monadic. The proof follows by composing the defining relationship θP ◦ ρ = σ ◦ Qθ with the inverse of θ on both sides.For the special case of the symmetric algebra monad in symmetric monoidal categories C and D , we obtain a monad functor from any symmetric monoidal functor. Lemma 1.7.
Let C and D be two symmetric monoidal categories and let F : C → D be asymmetric monoidal functor with structure morphisms θ X,Y : F ( X ) ⊗ D F ( Y ) → F ( X ⊗ C Y ) and ε : 1 D → F (1 C ) . The iterations θ X,...,X : F ( X ) ⊗ m → F ( X ⊗ m ) , together with ε at level , assemble into a natural transformation ρ : P D F → F P C . Then the pair ( F, ρ ) is a monad functor between the symmetric algebra monads on C and D .Proof. The proof is a straightforward verification, which can be done levelwise, using the decom-position of P ◦ in Proposition 1.3.By a similar construction, we also obtain a monadic transformation from a symmetric monoidaltransformation between symmetric monoidal functors.8 Power operations on ultra-commutative ring spectra
In this chapter, we give an introduction to power operations on the global homotopy groups of anultra-commutative ring spectrum, and construct external power operations for any orthogonalspectrum. We work throughout this article in the context of global homotopy theory, where theadjective ‘global’ indicates that we study equivariant spectra for all compact Lie groups at once.To do so, we use the framework of global orthogonal spectra provided by Stefan Schwede in [34],and we use this work as a general referencing point on the foundations of global homotopy theory.The notion of an orthogonal spectrum was first defined by Mandell, May, Schwede and Shipleyin [28], building on the idea of symmetric spectra, and the idea to study global phenomena wasalready present in [26]. A category of global orthogonal spectra was constructed by Bohmann in[8]. This category is equivalent to the one constructed by Schwede by [8, Theorem 6.2].This category is endowed with a symmetric monoidal structure, and commutative monoids arecalled ultra-commutative ring spectra. These spectra support power operations on their globalhomotopy groups. Such power operations are important additional structure, as emphasizedmost prominently in the work of Hill, Hopkins and Ravenel on the non-existence of elementsof Kervaire-invariant one in [19]. In Section 2.1, we recall the multiplicative aspects of globalhomotopy theory and the formalism of power operations.In Section 2.2, we study the requirements for a spectrum to have such operations in moredetail, and to this end we introduce external power operations in Construction 2.14. These aredefined on the homotopy groups of any orthogonal spectrum X , but only take values in thehomotopy groups of the symmetric powers P m X . For an ultra-commutative ring spectrum, thestructure morphism P X → X then recovers the usual power operations. We assume familiarity with the context of global orthogonal spectra and ultra-commutative ringspectra, as established in [34, Chapters 3-5]. We quickly collect the relevant notation.We denote by S p the category of orthogonal spectra. For any compact Lie group G andinteger k , we can associate to an orthogonal spectrum X its k -th G -equivariant homotopy group π Gk ( X ) [34, 3.1.11]. A morphism f : X → Y of orthogonal spectra is called a global equivalence ifit induces isomorphisms f ∗ : π Gk ( X ) → π Gk ( Y ) for all compact Lie groups G and all k . The result-ing homotopy category obtained by inverting global equivalences is called the global homotopycategory GH .The collection of homotopy groups π ( X ) = { π G ( X ) } G for any orthogonal spectrum X comesequipped with restriction maps α ∗ : π K ( X ) → π G ( X ) for any continuous homomorphism α : G → K of compact Lie groups [34, Construction 3.1.15], and with transfer maps tr GH : π H ( X ) → π G ( X )for any closed subgroup H ⊂ G [34, Construction 3.2.22]. These morphisms endow π ( X ) withthe structure of a global functor [34, Definition 4.2.2], and the category of global functors isdenoted GF . It is defined as the category of additive functors from the global Burnside category A [34, Construction 4.2.1] to abelian groups. Here the Burnside category has as objects the compactLie groups and the morphisms are generated by restrictions and transfers [34, Proposition 4.2.5].Composition inside A contains the information about the relations of compositions of transfersand restrictions, such as the double coset formula [34, Theorem 3.4.9].Note that the notion of a global functor is a global version of a Mackey functor for a fixedcompact Lie group G . The adjective ‘global’ refers to the fact that we allow restrictions alongarbitrary group homomorphisms, not just inclusions. There are various different global versionsof Mackey functors, and we refer to [34, Remark 4.2.16] for a discussion of the different definitions.9 emark . The global homotopy groups for a compact Lie group can be considered as a functor
GH → S ets, X π G ( X ) . This functor is representable by the suspension spectrum of the global classifying space B gl G by[34, Theorem 4.4.3 i)]. The orthogonal space B gl G is a generalization of the classical space BG ,and we recall its construction now:For two inner product spaces V and W , we denote with L ( V, W ) the space of linear isometricembeddings from V into W , and with L the resulting category of inner product spaces and linearisometric embeddings. An orthogonal space, according to [34, Definition 1.1.1], is a T -enrichedfunctor L → T from the category L into the category of topological spaces.Similarly, we denote by O ( V, W ) the Thom space of an orthogonal complement bundle over L ( V, W ), see [34, Construction 3.1.1] for details. Then, an orthogonal spectrum is a T -enrichedfunctor O → T ∗ into the category of based spaces.Let now G be a compact Lie group. We define, for any G -representation V , the orthogonal G -space L V = L ( V, _) with the right G -action by precomposition with the G -action on V , andthe orthogonal space L G,V = L ( V, _) /G . By [34, Proposition 1.1.26], the global homotopy typesof L V and L G,V are independent of the choice of the G -representation V as long as V is faithful.We then denote for any faithful V the orthogonal G -spaces E gl G = L V and B gl G = L G,V .We also recall the suspension spectrum of an orthogonal space X : We define for any innerproduct space V the value of the suspension spectrum as(Σ ∞ + X )( V ) = S V ∧ X ( V ) + , where the O ( V )-action on (Σ ∞ + X )( V ) is the diagonal action. The structure morphism for twoinner product spaces V and W is σ V,W : S V ∧ S W ∧ X ( W ) + → S V ⊕ W ∧ X ( V ⊕ W ) + , the smash-product of the homeomorphism S V ∧ S W ∼ = S V ⊕ W and the morphism X ( W ) → X ( V ⊕ W ) induced by the embedding W ֒ → V ⊕ W .There is also another description of the suspension spectrum of B gl G [34, 3.1.2]. Choose afaithful G -representation V , such that L G,V represents B gl G . Then, for any inner product space W , we define a homeomorphism S W ∧ L ( V, W ) + ∼ = O ( V, W ) ∧ S V , (2.2)called the untwisting isomorphism, using that we can trivialize the orthogonal complement bundleover L ( V, W ) with an additional copy of V . These homeomorphisms descend to G -orbits andassemble for varying W into an isomorphismΣ ∞ + B gl G → O ( V, _) ∧ G S V =: F G,V S V . (2.3)Moreover, we have a stable tautological class e G ∈ π G (Σ ∞ + B gl G ) as defined in [34, 4.1.12], andthe pair (Σ ∞ + B gl G, e G ) represents π G in the sense that GH (Σ ∞ + B gl G, X ) → π G ( X )[ f ] f ∗ ( e G ) (2.4)is a bijection for every orthogonal spectrum X .10e now give more details about the multiplicative structure on orthogonal spectra and ho-motopy groups.The category of orthogonal spectra has a symmetric monoidal structure using the smash productas defined in [34, Definition 3.5.1]. Using this symmetric monoidal structure, we can define a sym-metric algebra monad P X = W m ≥ P m X for orthogonal spectra X and a notion of commutativemonoids in this category. Definition 2.5.
An ultra-commutative ring spectrum is a commutative monoid in the category S p of orthogonal spectra. We write ucom for the category of ultra-commutative ring spectra andmultiplicative maps.Note that by Lemma 1.2, the categories of P -algebras and of ultra-commutative ring spectraare isomorphic.The most important property of these ultra-commutative ring spectra and the reason that wefavour them over the global analogue of E ∞ -spectra are equivariant power operations in thehomotopy groups. These arise from the strict multiplicative structure of the spectrum X . Assuch operations are also the main motivation to define the more general notion of G ∞ -ringspectra, we now give a recollection on power operations for ultra-commutative ring spectra. Remark . Both the categories Ab of abelian groups and the global Burnside category A have a symmetric monoidal structure, namely the tensor product ⊗ on Ab and the product × : A × A → A from [34, Theorem 4.2.15]. Using these, we define a symmetric monoidalstructure on the functor category GF as a Day convolution product [13]. This product is calledthe box product (cid:3) , and for an explicit construction we refer the reader to [34, Construction4.2.27]. Definition 2.7.
A global Green functor is a commutative monoid in the category GF endowedwith the symmetric monoidal structure provided by the box product. A morphism of globalGreen functors is a morphism of global functors compatible with the multiplication.Explicitly, the multiplication map R (cid:3) R → R of a global Green functor is equivalent bothto a family of multiplication maps × : R ( G ) × R ( K ) → R ( G × K )for all compact Lie groups G and K , and to a family of diagonal products · : R ( G ) × R ( G ) → R ( G ) (2.8)for all compact Lie groups G . These multiplications have to satisfy the properties explained after[34, Definition 5.1.3]. The relationship between these formulations via restrictions along diagonaland projections is elaborated upon in [34, Remark 4.2.20]. Proposition 2.9.
The equivariant homotopy groups of an ultra-commutative ring spectrum E define a global Green functor π ( E ) . This statement can be proven by considering the external multiplication map ⊠ : π G ( X ) × π K ( Y ) → π G × K ( X ∧ Y ) , (2.10)defined for any two orthogonal spectra X and Y in [34, Construction 4.1.20]. Postcomposingwith the multiplication map E ∧ E → E of the ultra-commutative ring spectrum E gives themultiplication × of the global functor π ( E ). This indeed defines the structure of a global Green11unctor by [34, Theorem 4.1.22] and the properties of the multiplication on E .Alternatively, we could also use the external diagonal product ⊡ : π G ( X ) × π G ( Y ) → π G ( X ∧ Y ) , defined in [34, Construction 3.5.12], and postcompose it with the induced map of the multipli-cation of E . Note that this product is called × in [34]. We choose to name it ⊡ instead to avoidconfusion with the multiplication map of a global Green functor and to highlight its connectionwith the diagonal product (2.8).As becomes apparent in the proof of Proposition 2.9, we do not need the full structure ofan ultra-commutative ring spectrum to prove that its homotopy groups form a global Greenfunctor, in particular, we do not need strict commutativity. All that is required is a morphism π ( µ ) which is unital, associative and commutative, so we could have only required that themorphism µ induces a commutative ring structure on X in the homotopy category. The reasonwe do not use this approach and instead assume strict commutativity is that this induces morestructure on the homotopy groups, namely power operations and the related norm maps. Remark . To define the power operations induced by an ultra-commutative ring spectrum,we recall the wreath product Σ m ≀ G of the symmetric group Σ m on m letters with a group G .We refer to [34, Construction 2.2.3] for details. The wreath product is defined as the semidirectproduct Σ m ≀ G = Σ m ⋉ G m with respect to the permutation action of the symmetric group onthe factors of G m . This has a natural action on the m -th power A m of a G -set A , given by the G -action on each factor and the permutation action of Σ m on the factors. In particular, if V is a G -representation, then V m is a Σ m ≀ G -representation, which is faithful if V is a non-zero faithful G -representation.Now we can define the power operations as follows: Let E be an ultra-commutative ringspectrum, then we define for f : S V → E ( V ) in π G ( E ), where V is some G -representation, the(Σ m ≀ G )-map P m ( f ) : S V m ∼ = ( S V ) ∧ m f m −−→ E ( V ) ∧ m µ V,...,V −−−−−→ E ( V m ) , (2.12)which represents an element in π Σ m ≀ G ( E ). The map µ V,...,V is the value of the m -fold multipli-cation map E ∧ m → E on the inner product space V m . This assignment defines a morphism P m : π G ( E ) → π Σ m ≀ G ( E )for all m ≥ Theorem 2.13.
For an ultra-commutative ring spectrum E , the global homotopy groups π ( E ) together with the operations P m : π G ( E ) → π Σ m ≀ G ( E ) as defined in (2.12) form a global power functor. For the proof we refer to [34, Theorem 5.1.11].12 .2 External power operations
After having set up the context of global power operations, we now ask the question whether weneed the full structure of an ultra-commutative ring spectrum to obtain these power operations.In classical homotopy theory, we have the notion of H ∞ -ring spectra from [12], which is onlydefined in the stable homotopy category. Such H ∞ -ring spectra also define power operations ontheir homotopy and cohomology groups. We define a global analogue in this paper. To facilitatethe proof that the global homotopy groups of a G ∞ -ring spectrum support power operations, weintroduce external power operations.The multiplication on the homotopy groups of a ring spectrum is generalized to an externalmultiplication as in (2.10), which exists for any two orthogonal spectra. It gives back the internalmultiplication pairing for a ring spectrum by composing with the multiplication map of the ringspectrum. We now generalize the notion of power operations to an external version that existsfor any orthogonal spectrum and gives back the internal power operations (2.12) after applyingthe iterated multiplication map µ V,...,V . Construction 2.14.
Let X be an orthogonal spectrum. We set P m X = X ∧ m / Σ m and definemaps ˆ P m : π G ( X ) → π Σ m ≀ G ( P m X )for every compact Lie group G and m ≥ G -representation V and a G -equivariant map f : S V → X ( V ) representing an element in π G ( X ), we setˆ P m ( f ) : S V m ∼ = ( S V ) ∧ m f m −−→ X ( V ) ∧ m i V,...,V −−−−→ X ∧ m ( V m ) pr( V m ) −−−−−→ ( P m X )( V m ) , where the morphism i V,...,V is the iteration of the universal bimorphism from the definition ofthe smash product (see [34, Definition 3.5.1]), and pr : X ∧ m → X ∧ m / Σ m is the projection.We claim that this map is Σ m ≀ G -equivariant: To see this, let( σ ; g ∗ ) = ( σ ; g , . . . , g m ) ∈ Σ m ≀ G be an element of the wreath product. Then we consider the following diagram, where σ · (_ )signifies the Σ m -action by permutation:( S V ) ∧ m X ( V ) ∧ m X ∧ m ( V m ) ( P m X )( V m )( S V ) ∧ m X ( V ) ∧ m X ∧ m ( V m ) ( P m X )( V m ) S σ · V m = σ · ( S V ) ∧ m σ · X ( V ) ∧ m ( σ · X ∧ m )( σ · V m ) ( P m X )( σ · V m ) f ∧ m g ∗ ( σ ; g ∗ ) i V,...,V g ∗ g ∗ g ∗ ( σ ; g ∗ ) f ∧ m σ i V,...,V σ σ σf ∧ m i V,...,V
In this diagram, the upper left square commutes by equivariance of f , the other squares onthe top commute as the horizontal map i V,...,V is an m -morphism of spectra, and pr also isa morphism of spectra. The squares on the bottom row commute by the symmetry propertyof both the direct sum of inner product spaces and the smash product of spectra, and as weexactly quotient out the permutation action on X ∧ m in the passage to P m X . Thus this diagramis commutative and proves that the morphism ˆ P m ( f ) is Σ m ≀ G -equivariant, hence defines anelement in π Σ m ≀ G ( P m X ).These maps ˆ P m are called external power operations. Note that the quotient X ∧ m → P m ( X ) is13ecessary for this definition, since on X ∧ m we have to consider the Σ m -action by permuting thefactors.These external operations fit into a commutative diagram π G ( X ) π G ( P X ) π G ( X ) π Σ m ≀ G ( P m X ) π Σ m ≀ G ( P X ) π Σ m ≀ G ( P m X ) , (incl ) ∗ ˆ P m (pr ) ∗ P m ˆ P m (incl m ) ∗ (pr m ) ∗ (2.15)where incl m : P m X → P X is the inclusion as the wedge summand indexed by m and pr m isthe projection onto the wedge summand indexed by m . Moreover, we use that P X as an ultra-commutative ring spectrum has power operations on its homotopy groups. This diagram exhibitsˆ P m as a retract of P m .These external operations satisfy similar properties to those of the internal operations P m as defined in [34, Definition 5.1.6], and they can be deduced from the internal relations on P X .To formulate these properties, we have to relate the different wreath products occurring in thedefinition of power operations. The following morphisms are also explained in [34, Construction2.2.3]. Remark . The wreath products are related via the following monomorphisms:For all i, j ≥
0, we have the morphismΦ i,j : (Σ i ≀ G ) × (Σ j ≀ G ) → Σ i + j ≀ G, (2.17)defined using juxtaposition of permutations. We will often drop the brackets in this expressionin the rest of this paper.Moreover, for all k, m ≥
0, we have the group homomorphismΨ k,m : Σ k ≀ (Σ m ≀ G ) → Σ km ≀ G, (2.18)defined via interpreting ( σ ; τ , . . . , τ k ) ∈ Σ k ≀ Σ m as a block permutation on { , . . . , k }×{ , . . . , m } and using the lexicographic ordering to obtain an element in Σ km .Lastly, the diagonal map ∆ : Σ m → Σ m × Σ m defines a diagonal map∆ m : Σ m ≀ ( G × H ) → Σ m ≀ G × Σ m ≀ H. (2.19)Note that the morphisms ϕ i,j : P i X ⊗ P j X → P i + j X and ψ k,m : P k P m X → P km X used in Proposition 1.3 are defined as the projections along the morphisms Φ i,j and Ψ k,m . Theyare used again in Proposition 2.20. In the same way, we also define a morphism δ m : P m ( X ∧ Y ) → P m ( X ) ∧ P m ( Y )as the projection along the monomorphism ∆ m . Proposition 2.20.
Let X be an orthogonal spectrum. Then the external power operations ˆ P m : π G ( X ) → π Σ m ≀ G ( P m X ) defined for all compact Lie groups G and all m ≥ satisfy thefollowing properties: ) ˆ P m (1) = 1 ∈ π Σ m ( S ) for the identity maps ∈ π e ( S ) and ∈ π Σ m ( S ) . Here, on the righthand side we use the identifications Σ m ≀ e ∼ = Σ m and P m S = S ∧ m / Σ m ∼ = S .ii) ˆ P = Id , under the identifications Σ ≀ G ∼ = G and P X ∼ = X .iii) For a continuous group homomorphism α : K → G , we have ˆ P m ◦ α ∗ = (Σ m ≀ α ) ∗ ◦ ˆ P m as maps π G ( X ) → π Σ m ≀ K ( P m X ) .iv) For two orthogonal spectra X and Y , and classes x ∈ π G ( X ) and y ∈ π G ( Y ) , we have ( δ m ) ∗ ( ˆ P m ( x ⊡ y )) = ˆ P m ( x ) ⊡ ˆ P m ( y ) in π Σ m ≀ G ( P m X ∧ P m Y ) , where δ m is the projection associated to the diagonal map ∆ : Σ m → Σ m × Σ m , explained after (2.19) .Equivalently, we have for x ∈ π G ( X ) and y ∈ π K ( Y ) the relation ( δ m ) ∗ ( ˆ P m ( x ⊠ y )) = ∆ ∗ m ( ˆ P m ( x ) ⊠ ˆ P m ( y )) in π Σ m ≀ ( G × K )0 ( P m X ∧ P m Y ) .v) For any compact Lie group G and i, j ≥ , we have Φ ∗ i,j ◦ ˆ P i + j = ( ϕ i,j ) ∗ ◦ ( ˆ P i ⊠ ˆ P j ) as maps π G ( X ) → π (Σ i ≀ G ) × (Σ j ≀ G )0 ( P i + j X ) , where Φ i,j is the homomorphism (2.17) , and ϕ i,j : P i X ∧ P j X → P i + j X is the projection along Φ i,j .vi) For any compact Lie group G and k, m ≥ , we have Ψ ∗ k,m ◦ ˆ P km = ( ψ k,m ) ∗ ◦ ˆ P k ◦ ˆ P m as maps π G ( X ) → π Σ k ≀ (Σ m ≀ G )0 ( P km X ) , where Ψ k,m is the homomorphism (2.18) , and ψ k,m : P k ( P m X ) → P km X is the projection along Ψ k,m .vii) For all compact Lie groups G , all m ≥ and all x, y ∈ π G ( X ) , we have ˆ P m ( x + y ) = X i + j = m tr i,j (( ϕ i,j ) ∗ ( ˆ P i ( x ) ⊠ ˆ P j ( y ))) in π Σ m ≀ G ( P m X ) , where tr i,j is the transfer along the monomorphism Φ i,j , and ˆ P ( x ) = 1 ∈ π e ( S ) is the multiplicative unit.viii) For every closed subgroup H ⊂ G of a compact Lie group G and for every m ≥ , we have ˆ P m ◦ tr GH = tr Σ m ≀ G Σ m ≀ H ◦ ˆ P m as maps π H ( X ) → π Σ m ≀ G ( P m X ) . x) The external power operations are natural in the orthogonal spectrum, i.e. for a morphism f : X → Y of orthogonal spectra, for a compact Lie group G and m ≥ , the square π G ( X ) π Σ m ≀ G ( P m X ) π G ( Y ) π Σ m ≀ G ( P m Y ) ˆ P m π G ( f ) π Σ m ≀ G ( P m f )ˆ P m commutes.Proof. We deduce the above properties form the fact that π ( P X ) is a global power functor, usingthat we can express the external power operations in terms of the internal power operations for P X via ˆ P m = (pr m ) ∗ ◦ P m ◦ η ∗ (2.15).The property i ) follows directly from the explicit definition in Construction 2.14. For ii ), wecan write ˆ P = (pr ) ∗ ◦ P ◦ η ∗ , and use that P = id for any global power functor and thatpr ◦ η = id holds. The properties iii ) and viii ) hold, since they hold for P m and both restrictionsand transfers are natural in the spectrum.For the multiplicativity property iv ), we consider the diagram π G ( X ) × π G ( Y ) π G ( X ∧ Y ) π G ( P X ) × π G ( P Y ) π G ( P X ∧ P Y ) π G ( P ( X ∧ Y )) π Σ m ≀ G ( P X ) × π Σ m ≀ G ( P Y ) π Σ m ≀ G ( P X ∧ P Y ) π Σ m ≀ G ( P ( X ∧ Y )) π Σ m ≀ G ( P m X ) × π Σ m ≀ G ( P m Y ) π Σ m ≀ G ( P m X ∧ P m Y ) π Σ m ≀ G ( P m ( X ∧ Y )) . η ∗ × η ∗ ⊡ η X ∧ Y ∗ ( η ∧ η ) ∗ P m × P m ⊡ P m P m δ ∗ pr ∗ × pr ∗ ⊡ (pr ∧ pr) ∗ pr ∗ δ ∗ ⊡ ( δ m ) ∗ This diagram commutes, since ⊡ is natural, the map δ : P ( X ∧ Y ) → P X ∧ P Y is a map ofultra-commutative ring spectra and ∧ is the coproduct in ucom, hence the power operations on π ( P ( X ∧ Y ) are determined by those on P X and P Y by P m ( x ⊡ y ) = P m ( x ) ⊡ P m ( y ).The multiplicativity relation( δ m ) ∗ ( ˆ P m ( x ⊠ y )) = ∆ ∗ m ( ˆ P m ( x ) ⊠ ˆ P m ( y ))follows from the one for ⊡ by rewriting the product ⊠ in terms of the product ⊡ .16or the property v ), we consider the diagram π G ( X ) π G ( P X ) π Σ i ≀ G ( P X ) × π Σ j ≀ G ( P X ) π Σ i ≀ G ( P i X ) × π Σ j ≀ G ( P j X ) π Σ i ≀ G × Σ j ≀ G ( P X ∧ P X ) π Σ i ≀ G × Σ j ≀ G ( P i X ∧ P j X ) π Σ i + j ≀ G ( P X ) π Σ i ≀ G × Σ j ≀ G ( P X ) π Σ i + j ≀ G ( P i + j X ) π Σ i ≀ G × Σ j ≀ G ( P i + j X ) . η ∗ ˆ P i + j ˆ P i × ˆ P j P i + j P i × P j × ⊠ (pr) ∗ ⊠ (pr i,j ) ∗ µ ∗ ( ϕ i,j ) ∗ (pr i + j ) ∗ Φ ∗ i,j (pr i + j ) ∗ Φ ∗ i,j This diagram commutes, where we use that the corresponding property for the internal operationsholds and that the multiplication of the ultra-commutative ring spectrum P X is defined on asummand P i X ∧ P j X of P X ∧ P X as ϕ i,j .For the property vi ), we consider the diagram π G ( X ) π G ( P X ) π Σ m ≀ G ( P X ) π Σ m ≀ G ( P m X ) π Σ km ≀ G ( P X ) π Σ k ≀ (Σ m ≀ G )0 ( P X ) π Σ k ≀ (Σ m ≀ G )0 ( P k P m X ) π Σ km ≀ G ( P km X ) π Σ k ≀ (Σ m ≀ X )0 ( P km X ) . η ∗ ˆ P km ˆ P m P km P m P k (pr m ) ∗ ˆ P k (pr km ) ∗ Ψ ∗ k,m (pr km ) ∗ ( ψ k,m ) ∗ Ψ ∗ k,m In this diagram, the top left square commutes by the properties of the power operations P • on P X , and the bottom left square commutes by naturality of the restriction Ψ ∗ k,m . The right squarecommutes by the calculation of the multiplication of the monad P , restricted to P m X , containedin the proof of Proposition 1.3. The calculation there shows that the k th iterated product on thesummand P m X lands in the summand P km X and that the corresponding map P k P m X µ −→ P km X is exactly ψ k,m . Since P k and ˆ P k differ exactly by the multiplication map, this proves that thisdiagram commutes.For the additivity property vii ), we do the following calculation, using that ˆ P m = (pr m ) ∗ ◦ P m ◦ η ∗ : ˆ P m ( x + y ) =(pr m ) ∗ ( P m ( η ∗ ( x + y )))= X i + j = m (pr m ) ∗ (cid:16) tr i,j (cid:0) P i ( η ∗ ( x )) × P j ( η ∗ ( y )) (cid:1)(cid:17) = X i + j = m tr i,j (cid:16) (pr m ) ∗ (cid:0) P i ( η ∗ ( x )) × P j ( η ∗ ( y )) (cid:1)(cid:17) X i + j = m tr i,j (cid:16) ( ϕ i,j ) ∗ (cid:0) (pr i ) ∗ ( P i ( η ∗ ( x ))) ⊠ (pr j ) ∗ ( P j ( η ∗ ( y ))) (cid:1)(cid:17) = X i + j = m tr i,j (cid:0) ( ϕ i,j ) ∗ ( ˆ P i ( x ) ⊠ ˆ P j ( y )) (cid:1) Here, in the fourth line, we used the right part of the diagram used for property v ) to rewritethe projection of the product.The naturality property ix ) follows from the fact that for any f : X → Y , the map P f : P X → P Y is a map of ultra-commutative ring spectra, and hence it induces a morphism of global powerfunctors on π . Moreover, f is clearly compatible with inclusions and projections. This finishesthe proof of the properties of the external power operations.In the above proof, we deduce the properties of the external power operations from thoseof the internal operations for the free ultra-commutative ring spectrum P X . If X already is anultra-commutative ring spectrum, we can also do the converse, obtaining power operations onthe homotopy groups from the external power operations: Proposition 2.21.
Let X be a P -algebra with structure morphism h = _ m ≥ h m : P X = _ m ≥ P m X → X. Then the maps P m = ( h m ) ∗ ◦ ˆ P m : π G ( X ) → π Σ m ≀ G ( X ) together with the unit map h : S → X define the structure of a global power functor on π ( X ) .Proof. By Proposition 1.3, the assumption that X is a P -algebra is equivalent to the followingproperties: i) h = Id under the identification P X ∼ = X . ii) For all i, j ≥
0, the diagram P i X ∧ P j X X ∧ X P X P i + j X X h i ∧ h j ϕ i,j pr h h i + j commutes. iii) For all k, m ≥
0, the diagram P k P m X P k X P km X X P k h m ψ k,m h k h km commutes.Using these compatibilities, the properties of the external power operations from Proposition2.20 directly translate into the properties from [34, Definition 5.1.6] needed to define a globalpower structure on π ( X ). For this translation, we use the naturality of transfer and restrictionmaps with respect to morphisms of spectra. 18n this proof, we see that we do not need the whole structure of an ultra-commutative ringspectrum to obtain the power structure on π ( X ). As the functor π : S p → GF factors overthe global homotopy category GH , we can define a class of structured ring spectra in the globalhomotopy category by specifying a suitably derived algebra structure in GH . This leads us to thedefinition of G ∞ -ring spectra, which support power operations on their global homotopy groups.19 G ∞ -ring spectra and their properties In this chapter, we give the definition of G ∞ -ring spectra in Definition 3.3. This notion isa homotopical version of structured ring spectra, with structure morphisms only defined in theglobal homotopy category. This structured multiplication allows us to construct power operationson the equivariant homotopy groups of a G ∞ -ring spectrum in Construction 3.6.The notion of G ∞ -ring spectra is a global generalization of the non-equivariant notion ofan H ∞ -ring spectrum from [12, Definition I.3.1]. In Section 3.2 we construct an adjunctionbetween G ∞ - and H ∞ -ring spectra. The left adjoint is a forgetful functor from the globallyequivariant G ∞ -ring spectrum to the non-equivariant H ∞ -ring spectrum. The right adjointexhibits a way to obtain a G ∞ -ring spectrum from an H ∞ -ring spectrum, thought of as a “globalBorel construction”. This also gives a way to generate examples of G ∞ -ring spectra which do notcome as the homotopy types of ultra-commutative ring spectra, see Remark 3.32. For this, weuse the non-equivariant examples of Noel in [30] and Lawson in [25] of H ∞ -ring spectra whichdo not rigidify to commutative ring spectra.In Section 3.3, we compare the derived symmetric powers to a global version of the extendedpowers D m X = ( E Σ m ) + ∧ Σ m X ∧ m in Theorem 3.37. This can be used to give an alternativedescription of G ∞ -ring spectra, which is closer to the original definition from [12]. G ∞ -ring spectra and their power operations We recall from Proposition 2.21 that we can use maps h m : P m X → X with certain properties,translating to the fact that X is a P -algebra, to define internal power operations on the homotopygroups of X from the external power operations defined in Construction 2.14. But all that isreally needed are such maps on the homotopy groups, hence we define the corresponding structureon the level of the global homotopy category GH . To do so, we make use of the positive globalmodel structures on S p and ucom, which are constructed in [34, Proposition 4.3.33 and Theorem5.4.3]. Lemma 3.1.
The functors S p ucom P U form a Quillen adjoint functor pair, with respect tothe positive global model structures on both sides.Proof. We already know that these functors are adjoint to one another. To prove that this isa Quillen adjunction, it suffices to show that the right adjoint U preserves both fibrations andacyclic fibrations. This is directly evident from the characterization of global equivalences andpositive global fibrations of ultra-commutative ring spectra by their underlying maps.In fact, the model structure on ucom is transferred from the positive model structure on S p along this adjunction.Now, every Quillen adjunction defines an adjunction on the homotopy categories, see [21,Lemma 1.3.10], hence we get an adjunction GH Ho(ucom) . L gl P Ho( U ) Note that U is already homotopical, so it can be derived without a fibrant replacement. Fromthis adjunction, we obtain the monad G = Ho( U ) ◦ L gl P : GH → GH . (3.2) Definition 3.3. A G ∞ -ring spectrum is an algebra over the monad G .20 xample 3.4. As we obtain the notion of G ∞ -ring spectra as algebras over a derived monad G = L gl P , we see that algebras over the point-set monad P , i.e. ultra-commutative ring spectra,also induce an G ∞ -ring structure on their global homotopy type. This already gives a broadclass of examples, which encompasses the sphere spectrum S , Eilenberg-Mac Lane spectra HF for a global power functor F as constructed in [34, Theorem 5.4.14], and the global versions ofThom and K -theory spectra from [34, Chapter 6].These examples as homotopy types of ultra-commutative ring spectra should, however, not pro-vide all G ∞ -ring spectra: If we only had G ∞ -ring spectra as homotopy types of ultra-commutativering spectra, we would not have needed the definition of G ∞ -ring spectra. In Theorems 3.33and 3.34, we provide examples of G ∞ -ring spectra which are not the homotopy type of an ultra-commutative ring spectrum such that the G ∞ -ring structure is induced by the ultra-commutativemultiplication. Remark . We also note that the definition of G ∞ -ring spectra is not the same as that of ahomotopy commutative ring spectrum in GH . This can be seen from the fact that a homotopycommutative ring spectrum does not support power operations on its homotopy groups, whereasProposition 3.8 proves the existence of power operations for G ∞ -ring spectra. Also note that G ∞ -ring spectra posses more structure than H ∞ -rings internal to the global homotopy category GH .This again can be seen using power operations, and the difference lies in the fact that the derivedsymmetric power G m X can be represented by the global extended power Σ ∞ + E gl Σ m ∧ Σ m X ∧ m as shown in Theorem 3.37. In contrast, for an H ∞ -ring spectrum, the non-equivariant extendedpowers Σ ∞ + E Σ m ∧ Σ m X ∧ m would be used. Construction 3.6.
We now construct the power operations on the equivariant homotopy groupsof a G ∞ -ring spectrum E . Since E is a G ∞ -ring spectrum, we have a map G E → E in thehomotopy category. The monad G is defined as Ho( U ) ◦ L gl P , so we can calculate G E as follows:Let q : QE → E be a positively cofibrant replacement of E . Then G E is represented by theorthogonal spectrum P ( QE ). Moreover, as the map q is a global equivalence, we only needto construct power operations for QE . The structure morphism ζ : P QE → QE in the globalhomotopy category defines a map π ( ζ ) : M m ≥ π ( P m QE ) ∼ = π ( P QE ) → π ( QE ) , where we use that π : GH → GF is additive on wedges, see [34, Corollary 3.1.37 (i)]. Then, wedefine P m = π ( ζ ) m ◦ ˆ P m : π G ( E ) ∼ = π G ( QE ) → π Σ m ≀ G ( P m QE ) → π Σ m ≀ G ( QE ) ∼ = π Σ m ≀ G ( E ) . (3.7) Proposition 3.8.
Let E be a G ∞ -ring spectrum with structure map ζ : G E → E . Then theoperations P m defined in (3.7) , together with the multiplication given by ζ , define a structure ofa global power functor on π ( E ) .Proof. The components π ( ζ ) m : π ( P m QE ) → π ( QE ) satisfy the compatibility diagrams fromProposition 2.21, as ζ defines an algebra structure over G . Hence, the definition P m = π ( ζ ) m ◦ ˆ P m defines internal power operations on the homotopy groups of E .Moreover, π ( QE ) obtains the structure of a global Green functor from the component maps ζ : P QE → QE , which defines a homotopy commutative multiplication on the homotopy typeof E , and ζ : S → QE , which is the unit of the homotopy multiplication. Hence, the globalfunctor π ( E ) has the structure of a global power functor.21 .1.1 External power operations via representability Now, we have defined G ∞ -ring spectra and shown that their homotopy groups are equippedwith power operations. We constructed these power operations using external power operations,defined by a point-set construction. The definition of G ∞ -ring spectra however is, up to thedefinition of the monad G , internal to the global homotopy category GH . Hence it would beconvenient to also define the power operations ˆ P m internal to the global homotopy category. Thiscan be done by deriving the levels of P separately and using representability of the homotopygroups π G by Σ ∞ + B gl G . For this, we again use the positive global model structures from [34,4.3.33, 5.4.3]. Lemma 3.9.
Let f : X → Y be a global equivalence between positively cofibrant spectra, and let m ≥ . Then P m f : P m X → P m Y is a global equivalence in S p .Proof. We follow the argument given at the end of the proof of [34, Theorem 5.4.12]. By 3.1,we know that the functor P : S p → ucom is left Quillen. By Ken Brown’s lemma [21, Lemma1.1.12], P f : P X → P Y is a weak equivalence of ultra-commutative ring spectra, hence by def-inition a weak equivalence of the underlying spectra. But the transformations P m incl m −−−→ P = W m ≥ P m pr m −−→ P m exhibit P m f as a retract of P f , thus also this morphism is a global equiva-lence. Remark . This lemma is enough to conclude that P m : S p → S p admits a left derived functor G m : GH → GH . However, one can indeed show more: for any m >
0, the functor P m : S p → S p preserves positivecofibrations and acyclic positive cofibrations between positively cofibrant spectra. This uses[17, Theorem 22], that the positive cofibrations are symmetrizable [34, Theorem 5.4.1] and thatpositively cofibrant spectra are flat [34, Theorem 4.3.27].We now calculate the value of the functor G on the sphere spectrum and on the representingspectra Σ ∞ + B gl G for the global homotopy groups. We use this to define the external poweroperations intrinsically in GH . For this, we use the decomposition G = W m ≥ G m provided bythe above lemma. Example 3.11.
We calculate the value of G on the sphere spectrum S :Since S is not positively cofibrant, we need to positively replace it. For this, consider the map λ Σ , R , : F Σ , R S → F Σ , = S from [34, 4.1.28]. This map is a global equivalence by [34, Theorem 4.1.29], as 0 is a faithfulrepresentation of the trivial group Σ . Moreover, the spectrum F Σ , R S = O ( R , _) ∧ S ispositively cofibrant, hence this map λ Σ , R , can be chosen as a positively cofibrant replacement.Then, we have G m S ∼ = P m ( F Σ , R S ) ∼ = O ( R m , _) ∧ Σ m S m = F Σ m , R m S m . Here, we used that we can compute the product of semifree orthogonal spectra F G,V A and F K,W B as F G × K,V ⊕ W ( A ∧ B ) by [34, 4.1.26]. Hence, by the description of the global classifyingspaces via semifree orthogonal spectra in (2.3), we see that G m S ∼ = Σ ∞ + B gl Σ m . G m (Σ ∞ + B gl G ) for any compact Lie group, with the previouscalculation a special case for G = e , using the identification Σ ∞ + B gl e ∼ = S . For this calculation,choose a non-zero faithful G -representation V . Then we can writeΣ ∞ + B gl G ∼ = F G,V S V as in (2.3). Now, the spectrum F G,V S V is positively cofibrant, and hence we obtain G m ( F G,V S V ) = P m ( F G,V S V ) ∼ = F G m ,V m S V m / Σ m ∼ = F Σ m ≀ G,V m S V m , where for the last identification, we used that the permutation action of Σ m and the action of G m on V m assemble into the natural action of Σ m ≀ G . Then V m is a faithful Σ m ≀ G -representation,hence we see that G m (Σ ∞ + B gl G ) ∼ = Σ ∞ + B gl (Σ m ≀ G ) . (3.12) Construction 3.13.
We now give another description of the external power operation, usingour calculation of G m on the representing spectra for π G .Let f ∈ π G ( X ) be an element of the homotopy groups of a global homotopy type X . Bythe representability result (2.4), we can represent f by a map f : Σ ∞ + B gl G → X in the globalhomotopy category. Then, we define for m ≥ G m ( f ) : Σ ∞ + B gl (Σ m ≀ G ) ∼ = G m (Σ ∞ + B gl G ) G m f −−−→ G m X, and this morphism represents an element in π Σ m ≀ G ( G m X ). Thus we define the external opera-tions as the effect of the functor G m on the homotopy groups π G ( X ) ∼ = GH (Σ ∞ + B gl G, X ), andobtain maps G m : π G ( X ) → π Σ m ≀ G ( G m X ) . Lemma 3.14.
For a positively cofibrant spectrum X , the two external operations ˆ P m and G m : π G ( X ) → π Σ m ≀ G ( P m X ) agree.Proof. Let f ∈ GH (Σ ∞ + B gl G, X ), then the corresponding class in π G ( X ) is f ∗ ( e G ). Concretely,let f : S V ∧ ( B gl G ) + ( V ) → X ( V ) for a non-zero faithful G -representation V . We can alwaysrepresent f on a faithful G -representation, since we can embed any G -representation into afaithful one. Then, the tautological class in π G (Σ ∞ + B gl G ) is e G : S V _ ∧ Id V −−−−→ S V ∧ ( B gl G ) + ( V ) = S V ∧ L ( V, V ) + /G, and the tautological class for Σ m ≀ G is e Σ m ≀ G : S V m _ ∧ Id V m −−−−−→ S V m ∧ L ( V m , V m ) + / (Σ m ≀ G ) . Note that this element agrees withˆ P m e G : S V m (_ ∧ Id V ) m −−−−−−−→ S V m ∧ ( L ( V, V ) + /G ) m pr −→ S V m ∧ L ( V m , V m ) + / (Σ m ≀ G ) . We now compare ˆ P m ( f ∗ ( e G )) and ( G m f ) ∗ ( e Σ m ≀ G ). Since X is positively cofibrant, we can write P m instead of G m . Then by naturality of the external power operations, we obtainˆ P m ( f ∗ ( e G )) = P m ( f ) ∗ ( ˆ P m ( e G )) = P m ( f ) ∗ ( e Σ m ≀ G ) . Hence, the two operations agree. 23hen, for a G ∞ -ring spectrum X with structure maps ζ m : G m X → X , we can define poweroperations on its homotopy groups by P m = ( ζ m ) ∗ ◦ G m : π G ( X ) → π Σ m ≀ G ( X ) . It is clear from the above lemma that this definition agrees with the one given in Construction3.6, since they agree for positively cofibrant representatives of the spectrum X .This definition has the advantage that it only lives in the global homotopy category and makesno direct reference to the positive global model structure. We will use this description whenanalysing whether global Moore spectra have the structure of G ∞ -ring spectra. Remark . As an application of this description of the external power operations, we also defineexternal cohomology operations, and show that a G ∞ -structure can be used to internalize theseoperations. These internal cohomology operations are also constructed for an ultra-commutativering spectrum in [34, Remark 5.1.14].Let X be an orthogonal spectrum and A be a cofibrant based G -space. Then we define the G -equivariant X -cohomology of A as X G ( A ) = [Σ ∞ + L G,V
A, X ] , where [_ , _] denotes the morphisms in GH , and V is any faithful G -representation. Then,external power operations on this X -cohomology are defined byˆ P m : X G ( A ) = [Σ ∞ + L G,V
A, X ] G m −−→ [ G m Σ ∞ + L G,V A, G m X ]=[Σ ∞ + L Σ m ≀ G,V m A m , G m X ] = ( G m X ) m ≀ G ( A m ) . Here, we used a relative version of the calculations in 3.11 to calculate G m Σ ∞ + L G,V A ∼ = Σ ∞ + L Σ m ≀ G,V m A m . Using a G ∞ -ring structure on X , given by morphisms ζ m : G m X → X , we can internalize theseoperations to P m : X G ( A ) ˆ P m −−→ ( G m X ) m ≀ G ( A m ) ( ζ m ) ∗ −−−−→ X m ≀ G ( A m ) . In [34, Remark 5.1.14], it is shown that these power operations forget to the classical power oper-ations X ( A ) → X ( B Σ m × A ) on the non-equivariant X -cohomology of A upon postcompositionwith the diagonal on A . G ∞ - and H ∞ -ring spectra In this section, we compare the notion of G ∞ -ring spectra to the classical notion of H ∞ -ringspectra. This is accomplished by lifting the adjunction GH SH UR to structured ring spectra, where U is the forgetful functor and R its right adjoint. This isexhibited in [34, Theorem 4.5.1]. We start with the forgetful functor.24 .2.1 Lifting the forgetful functor GH → SH to structured ring spectra
We first recall the classical definition of H ∞ -ring spectra: As defined in [12, I, Definition 3.1], an H ∞ -ring spectrum X is defined by maps ξ m : D m X → X, where D m X = ( E Σ m ) + ∧ Σ m X ∧ m . These maps are required to satisfy compatibility conditionsas they are also mentioned in Proposition 1.3. Note that this formulation uses the modern smashproduct, which was not yet available in the original definition. Unravelling the definitions in [12]however gives this formulation. In contrast, our definition of G ∞ -ring spectra uses a modernpoint-set category of spectra to obtain the monad G , and defines G ∞ -ring spectra as algebrasover this monad. As this definition is more conceptual and allows us to use the results fromSection A.1, we also formulate the notion of H ∞ -ring spectra in this way.For this, note that the adjunction S p Com P U Com is also a Quillen adjunction with respect to the positive stable model structures defined by Stolzin [37, Proposition 1.3.10 and Theorem 1.3.28]. Thus we obtain a derived adjunction SH Ho st (Com) . L st P Ho( U Com ) Definition 3.16. An H ∞ -ring spectrum is an algebra over the monad H = Ho( U Com ) ◦ L st P .By abuse of notation, we will also denote H as L st P , since it is the left derived functor of P : S p → S p . In the same way, we denote G by L gl P .That our definition using L st P agrees with the original definition follows from the followingstatement, after the necessary translations regarding the different models for spectra: Lemma 3.17.
Let X be a positive stably cofibrant orthogonal spectrum. Then the map p : D m X = ( E Σ m ) + ∧ Σ m X ∧ m → X ∧ m / Σ m = P m X that collapses E Σ m is a stable weak equivalence.Proof. This is the statement of [37, Lemma 1.3.17], where a cellular induction along the lines of[12, p. 36-37] is carried out.Using this definition of H ∞ -ring spectra, we show that the underlying stable homotopy typeof a G ∞ -ring spectrum is an H ∞ -spectrum. To do this, we show that the derived functor U : GH → SH is a monad functor in the sense of 1.4. We deduce this formally from a variant ofthe fact that taking the homotopy category of a model category is a pseudo-2-functor [21, 1.4.2f]:We consider the 2-category (Model , left) of model categories and left Quillen functors. Then,[21, 1.4.3] shows that taking homotopy categories and left derived functors is a pseudo 2-functor L : (Model , left) → Cat. Then, by Corollary A.5 the functor L preserves monads and monadmorphisms.However, the functor P : S p → S p is not left Quillen, but merely left derivable, i.e. it sendsweak equivalences between cofibrant objects to weak equivalences, in both the stable and theglobal positive model structure. Moreover, all compositions P ◦ k : S p → S p can be derived:25 emma 3.18. Let X be a positively cofibrant spectrum in either the stable or global modelstructure, and let A = W I S be a wedge of sphere spectra. Then P ( A ∨ X ) ∼ = B ∨ Y , where B = W J S is a wedge of spheres which only depends on A , and where Y is a positively cofibrantspectrum. Moreover, if f : X → X ′ is a weak equivalence between positively cofibrant spectra,then also P ( id ∨ f ) is a weak equivalence of the form id ∨ g : B ∨ Y → B ∨ Y ′ .In particular, for any k ≥ , the functor P ◦ k : S p → S p sends weak equivalences between positivelycofibrant spectra to weak equivalences.Proof. We write P ( A ∨ X ) ∼ = P ( A ) ∧ P ( X ) ∼ = P ( A ) ∧ ( S ∨ P > ( X )) ∼ = P ( A ) ∨ ( P ( A ) ∧ P > ( X ))Now, we see that P ( A ) = P _ I S ! ∼ = ^ I ( PS ) ∼ = ^ I _ i ≥ S ∼ = _ N I ^ I S ! ∼ = _ N I S is a wedge of spheres. Moreover, the spectrum P > X is positively cofibrant by applying [17,Corollary 10] to the positive model structures, and hence also P ( A ) ∧ P > X is positively cofibrant.This proves the first assertion, putting B = P ( A ) and Y = P ( A ) ∧ P > X . If f : X → X ′ is aweak equivalence between positively cofibrant spectra, so are P > ( f ) and P ( A ) ∧ P > ( f ). Thisproves the second part of the lemma, since P ( id ∨ f ) = id P ( A ) ∨ ( P ( A ) ∧ P > ( f )).In total, this proves the conclusion that P ◦ k preserves weak equivalences between positivelycofibrant spectra by induction. Remark . Note that this lemma uses the results from [17] to prove that P > preservescofibrancy, and hence depends on the fact that cofibrations in the positive model structures from[34, 37] are symmetrizable. Moreover, it uses that positive cofibrant spectra are flat. This isthe reason we use the positive stable model structure defined by Stolz, since it has exactly theseproperties.Now we generalize the statement of [21, 1.4.3] to encompass all left derivable functors. Thereare two problems: the class of left derivable functors is not closed under composition, and if F and G are composable left derivable functors such that GF also is left derivable, the naturaltransformation LG ◦ LF → L ( GF ) might not be invertible. However, we obtain the followingresult: Proposition 3.20.
Let (Model , all) be the -category of model categories and all functors andnatural transformations, and let LD er denote the class of all left derivable functors and LD er the class of all natural transformations between left derivable functors. Then the assignment L : (Model , LD er , LD er ) → Cat
C 7→
Ho( C ) , F LF, η Lη comes equipped with the following structure:i) A unitality isomorphism α C : id Ho( C ) → L ( id C ) for any model category C .ii) A natural transformation µ G,F : LG ◦ LF → L ( GF ) for any pair of left derivable functors F : C → D , G : D → E such that GF is also left derivable.These satisfy the properties of a lax -functor from Definition A.2 where they are defined.Moreover, if F : C → D is left derivable and U : D → E is homotopical, then
U F is left derivableand µ U,F is invertible. roof. The proof is the same as that of [21, 1.4.3], where we weaken the requirement of beingleft Quillen to sending weak equivalences between cofibrant objects to weak equivalences.We consider the following commutative diagram S p gl S p st S p gl S p st , U P gl P st U (3.21)which exhibits the functor U as a monad functor between ( P gl , µ gl , η gl ) and ( P st , µ st , η st ). More-over, since U is homotopical, it guarantees that all composites P ◦ i ◦ U ◦ P ◦ j = U ◦ P ◦ i + j areleft derivable. Proposition 3.20 allows us to conclude that taking homotopy categories and leftderived functors preserves the monads P on S p gl and S p st as well as the functor U between them. Proposition 3.22.
The left derived functors L P st and L P gl have the structure of monads viathe natural transformations Lµ gl ◦ µ P gl , P gl , Lη gl ◦ α S p gl and the analogous transformations for L P st .Moreover, the derived functor Ho( U ) : Ho( S p gl ) → Ho( S p st ) has the structure of a monad functorbetween L P gl and L P st via the transformation µ − U, P gl ◦ µ P st ,U .Proof. This is the statement of Corollary A.5. To apply this corollary as stated, we would needto have a lax 2-functor L : Model → Cat encompassing all left derivable functors. However, itsuffices that we have the required structure morphisms for all composites P ◦ i st ◦ U ◦ P ◦ j gl with i + j ≤
3. This is the case by Lemma 3.18 and the commutative square 3.21.Thus, we have proven the following:
Proposition 3.23.
Let X be a G ∞ -ring spectrum with structure map h : L P gl X → X . Then themap h ◦ µ − U, P gl ◦ µ P st ,U : ( L st P ) U X → U X defines an H ∞ -ring structure on the stable homotopytype U X . Moreover, for a G ∞ -ring morphism f : X → Y in GH between two G ∞ -ring spectra,the map U ( f ) ∈ SH ( U X, U Y ) is an H ∞ -ring map.Thus the functor U lifts to a functor from the category of G ∞ -ring spectra to the category of H ∞ -ring spectra. Using this result, we get the commutative diagramHo gl (ucom) ( G ∞ -ring spectra)Ho st (Com) ( H ∞ -ring spectra) (3.24)of homotopy categories of structured ring spectra, where all functors are forgetful ones. SH → GH to structured ring spectra
In this section, we study whether the forgetful functor U from the category of G ∞ -ring spectrato H ∞ -ring spectra from (3.24) has adjoints. The corresponding question for the homotopycategories SH and GH is investigated in [34, Chapter 4.5], and we use these results to obtaina right adjoint to the forgetful functor. This gives us a way to define G ∞ -ring spectra from27 ∞ -ring spectra, and we use this to give examples of G ∞ -ring spectra which do not come fromultra-commutative ring spectra.We first recall the right adjoint R : SH → GH to the forgetful functor from [34, Construction4.5.21].
Construction 3.25.
We define the functor b : S p → S p as a “global Borel construction”: For an orthogonal spectrum X and an inner product space V ,we set ( bX )( V ) = map( L ( V, R ∞ ) , X ( V )) , with structure morphisms defined as in [34, 4.5.21].We also define a natural transformation i : Id → b via the map i X ( V ) : X ( V ) → map( L ( V, R ∞ ) , X ( V )) , x const x . The morphism i X ( V ) is a non-equivariant homotopy equivalence for all inner product spaces V ,as the space L ( V, R ∞ ) is contractible. Hence the induced morphism i X : X → bX is invertiblein the stable homotopy category.Moreover, b comes equipped with a lax symmetric monoidal structure for the smash product oforthogonal spectra, such that i : Id → b is a monoidal transformation. Then, by Lemma 1.7, weobtain the following: Corollary 3.26.
The functor b : S p → S p defines a monad endofunctor in the sense of (1.4) ofthe symmetric algebra monad P on S p .Moreover, the transformation i : Id → b is a monadic transformation. By [34, Propositon 4.5.22], the functor b represents the right adjoint to the forgetful functor U : GH → SH on stable Ω-spectra. We modify this statement to hold on positive Ω-spectra,since we need positive model structures for the study of commutative ring spectra. Note thatthe positive stable model structure from [37, Proposition 1.3.10] has fewer fibrant objects thanthe projective model structure, so all fibrant objects are in particular positive Ω-spectra.
Proposition 3.27.
Let X be a positive orthogonal Ω -spectrum.i) Then bX is a positive global Ω -spectrum whose homotopy type lies in the image of the rightadjoint R .ii) For every orthogonal spectrum A , the two homomorphisms GH ( A, bX ) U −→ SH ( A, bX ) ( i X ) − ∗ −−−−→ SH ( A, X ) are isomorphisms.Proof. The proof is completely analogous to the proof of [34, Proposition 4.5.22]. The cited proofthat bX is a global Ω-spectrum works level-wise, hence if X is only a positive Ω-spectrum, bX is a positive global Ω-spectrum. That bX is right induced from the stable homotopy categorycan be seen by replacing bX by the globally equivalent Ω sh( bX ) and using that the shift of apositive Ω-spectrum is a Ω-spectrum.Moreover, ii ) is a formal consequence of i ), as indicated in the proof of [34, 4.5.22].28e can use this point-set level lift together with the positive global model structure to answerthe question whether the functor R extends to a right adjoint of the forgetful functor from G ∞ - to H ∞ -ring spectra. For this, we derive the monad functor b and the monadic transformation i withrespect to the non-equivariant and global positive model structures. To do this, we first need tocheck that b can be derived to a functor on the homotopy categories. Here, we again use that thefibrant objects in the convenient stable model structure are in particular Ω-spectra. We provethe following proposition in this more general context, since the forgetful functor U : S p gl → S p st sends global Ω-spectra to non-equivariant Ω-spectra. Lemma 3.28.
Let f : X → Y be a stable equivalence between (positive) Ω -spectra. Then b ( f ) isa global equivalence.Proof. Since f is a stable equivalence between (positive) Ω-spectra, it is a (positive) level equiv-alence. Since the G -space L ( V, R ∞ ) is G -cofibrant and free for any faithful G -representation V ,mapping out of it takes weak equivalences to G -weak equivalences. Thus, b takes (positive) levelequivalences to (positive) strong level equivalences and thus to global equivalences. Hence, b ( f )is a global equivalence.This lemma tells us that the functor b can be right-derived to define a functor R = Rb : SH →GH . This allows us to describe the unit of the adjunction
GH SH UR in a similar way to theresults for the counit in Proposition 3.27 ii ). Lemma 3.29.
Let X and Y be (positive) Ω -spectra, and assume that X is moreover cofibrant.Then, the composition SH ( X, Y ) b −→ GH ( bX, bY ) ( i X ) ∗ −−−→ GH ( X, bY ) is a bijection inverse to GH ( X, bY ) U −→ SH ( X, bY ) ( i Y ) − ∗ −−−−→ SH ( X, Y ) . Proof.
We consider the diagram GH ( X, bY ) GH ( bX, bY ) SH ( X, bY ) SH ( X, Y ) . U ( i X ) ∗ ( i Y ) ∗ b As both ( i Y ) − ∗ and U are bijective, it suffices to show that this diagram is commutative, i.e.that ( i Y ) ∗ = U ◦ ( i X ) ∗ ◦ b .Let f ∈ SH ( X, Y ), we can represent it by a map f : X → Y of spectra by the assumptions on X and Y . Then for any inner product space V , we have (cid:0) ( i Y ) ∗ ( f ) (cid:1) ( V ) = ( i Y ◦ f )( V ) : X ( V ) → bY ( V ) = map( L ( V, R ∞ ) , Y ( V )) x const f ( V )( x ) , where const f ( V )( x ) is the constant map at the value f ( V )( x ). Moreover, the other compositionaround the diagram sends f to the map ( U ◦ ( i X ) ∗ ◦ b )( f ), which at the level of an inner productspace V has the form( bf ◦ i X )( V ) : X ( V ) → bY ( V ) = map( L ( V, R ∞ ) , Y ( V ))29 ( bf ( V ))(const x ) = f ( V ) ◦ const x = const f ( V )( x ) Hence the above square commutes, and ( i X ) ∗ ◦ b is the inverse to ( i Y ) − ∗ ◦ U .Now, we set up the double-categoric context we use to prove that the above adjunction liftsto G ∞ - and H ∞ -ring spectra.We first consider the double category Model of model categories, left Quillen functors as verticalmorphisms, right Quillen functors as horizontal morphisms and all natural transformations as2-cells. Then, [35, Theorem 7.6] shows that taking the homotopy category and derived functorsdefines a pseudo double functor into the double category Sq (Cat) of categories, functors ashorizontal and vertical morphisms and natural transformations as 2-cells.In our context, however, neither the symmetric algebra monads P gl and P st nor b are Quillenfunctors, but merely derivable. Hence, as in Proposition 3.20, we restrict to the classes of leftand right derivable functors respectively, and obtain the following result: Proposition 3.30.
Let ( Model , all , all) denote the double category of model categories and allfunctors as horizontal and vertical morphisms, and natural transformations as 2-cells. Let LD er be the class of left derivable functors, RD er denote the class of right derivable functors and D er denote the class of natural transformations of the form F G → KH with G, K right derivableand
F, H left derivable. Then the assignment
Ho : (
Model , LD er, RD er, D er ) → Sq (Cat) C 7→
Ho( C ) , F LF, G RG, η Ho( η ) comes equipped with the following structure:i) Unitality isomorphisms α v C : id Ho( C ) → L ( id C ) and α h C : R ( id C ) → id Ho( C ) for any modelcategory C .ii) A natural transformation µ vG,F : LG ◦ LF → L ( GF ) for any pair of left derivable functors F : C → D , G : D → E such that GF is also left derivable.iii) A natural transformation µ hG,F : R ( GF ) → RG ◦ RF for any pair of right derivable functors F : C → D , G : D → E such that GF is also right derivable.These satisfy the properties of a lax-oplax double functor from Definition A.6 where they aredefined.Moreover, if F : C → D and G : D → E are right derivable and either F is right Quillen or G ishomotopical, then GF is right derivable and µ hG,F is invertible.Proof. The proof is the same as for [35, Theorem 7.6], where we weaken the requirements frombeing Quillen to being derivable. The last statement about invertibility of µ hG,F follows fromthe description of this transformation as ( GF ) P Gp FP −−−−→ GP F P , where p : id → P denotes afunctorial fibrant replacement. If G is homotopical, it sends the weak equivalence p F P to a weakequivalence. If F is right Quillen, F P is fibrant and thus p F P is a weak equivalence betweenfibrant objects. Since G is right derivable, it then sends p F P to a weak equivalence.Now, we have all the ingredients to prove that we have an adjunction between G ∞ -ringspectra and H ∞ -ring spectra. Theorem 3.31.
Let R : SH → GH denote the right adjoint to U : GH → SH . ) The functor R : SH → GH induces a functor ˆ R from the category of H ∞ -ring spectra to thecategory of G ∞ -ring spectra.ii) The functor ˆ R is right adjoint to the forgetful functor U , with adjunction unit lifted from I : id GH → RU and adjunction counit lifted from J − : U R → id SH , where both I and J areobtained from deriving i : id → b .Proof. i) We have seen that P ◦ i gl and P ◦ j st are left derivable for all i, j ≥ b is rightderivable, and moreover that b has the structure of a monad morphism between P st and P gl by Corollary 3.26. Hence the above Proposition 3.30 suffices to invoke Proposition A.10 toconclude that the right derived functor R = Rb : SH → GH is a monad functor between L st P and L gl P . Hence, it lifts to a functor ˆ R : ( H ∞ -Rings) → ( G ∞ -Rings). ii) We know that both R = R ( b ) and U = R ( u ) for the forgetful functor u : S p gl → S p st aremonad functors. Hence both compositions RU and U R are monad functors. We show that i : id S p → b defines monadic transformations I : id GH → RU and J − : U R → id SH , so thatwe obtain lifts to the categories of algebras.First we note that both compositions ub and bu are right derivable, since u homotopical andsends global Ω-spectra to non-equivariant Ω-spectra, on which b is homotopical by Lemma3.28. Moreover, they are monad functors as composites of monad functors, and hence so arethe derived functors R ( ub ) and R ( bu ). We start by constructing I .We define I = µ hb,u ◦ Ho( i ) ◦ ( α h S p gl ) − : id GH → R ( id S p gl ) → R ( bu ) → Rb ◦ Ru = RU.
Since i is a monadic transformation by Corollary 3.26 and any lax-oplax double functorpreserves these by Proposition A.10, we see that Ho( i ) is monadic. Moreover, by LemmaA.11, both µ hb,u and α h S p gl are monadic, and thus also ( α h S p gl ) − . In total, I is a monadictransformation as a composition of such.Analogously, we define J = µ hu,b ◦ Ho( i ) ◦ ( α h S p st ) − , this also is a monadic transformation bythe same arguments. Moreover, J is invertible, since µ hu,b is by Proposition 3.30, ( α h S p st ) − by definition and Ho( i ) is invertible in the stable homotopy category since i is a stableequivalence. Thus also J − : U R → id SH is a monadic transformation.Hence, the two transformations I : id GH → RU and J − : U R → id SH lift to the categories ofalgebras. Moreover, Lemma 3.29 shows that for any homotopy types X ∈ GH and Y ∈ SH ,the morphisms SH ( U X, Y ) R −→ GH ( RU X, RY ) I ∗ −→ GH ( X, RY )and GH ( X, RY ) U −→ SH ( U X, U RY ) J − ∗ −−→ SH ( U X, Y )are inverse isomorphisms, and thus I and J − are unit and counit of an adjunction. Asthe forgetful functor from the categories of algebras to the base category is faithful, thisproperty lifts to prove that the lifts of I and J − are unit and counit of an adjunction G ∞ -Rings H ∞ -Rings U ˆ R . As an application of this result, we use the right adjoint to give examples of G ∞ -ring spectrawhich are not obtained as the homotopy type of an ultra-commutative ring spectrum.31 emark . Let X ∈ SH be an H ∞ -ring spectrum. Then we consider the induced G ∞ -ringstructure on the global homotopy type RX . This induces an H ∞ -ring structure on U ( RX ). Themap J X : X → U ( RX ), defined in the proof of Theorem 3.31, is a stable equivalence and bymonadicity of the transformation in isomorphism of H ∞ -ring spectra.Assume now that there is an ultra-commutative ring spectrum Y such that the homotopy typeof Y is RX , and such that the G ∞ -ring structure is induced from the structure map P Y → Y of Y . Then, the H ∞ -ring structure on U ( RX ) is induced by the commutative multiplication on U Y .But the H ∞ -ring spectrum U ( RX ) is equivalent to X . Hence, if we find an ultra-commutativerepresentative Y for the G ∞ -ring spectrum RX , then U Y is a commutative ring spectrum whichinduces the H ∞ -ring structure on X .Thus, if we want to provide examples of G ∞ -ring spectra that are not induced by ultra-commutative ring spectra, it is enough to consider this question non-equivariantly, where coun-terexamples are already exhibited in the papers [30] and [25]. We summarize these constructions:In his paper [30], Justin Noel uses a counterexample to the transfer conjecture constructedby D. Kraines and T. Lada in [24]. The transfer conjecture roughly states that there is anequivalence between the homotopy category of infinite loop spaces and the homotopy categoryof spaces admitting certain transfers. Generalizing the notions of Q n -spaces of Kraines and Ladato spectra, Noel translates their notion of a Q -space into that of an H ∞ -ring spectrum, and thenotion of an Q ∞ -space into that of an E ∞ -ring spectrum. Then, he uses [24, Theorems 8.1 and8.2] to obtain the following theorem as [30, Theorem 1.2]: Theorem 3.33.
Let s k ∈ H k ( BU ; Z (2) ) be a primitive generator. Define the space KL k as thehomotopy fibre KL k i k −→ BU (2) 4 s k −−→ K ( Z (2) , k ) and consider the suspension spectra Σ ∞ + KL k Σ ∞ + i k −−−→ Σ ∞ + BU (2) Σ ∞ + s k −−−−→ Σ ∞ + K ( Z (2) , k ) Then, for any k , the spectrum Σ ∞ + KL k admits the structure of an H ∞ -ring spectrum, and Σ ∞ + i k is an H ∞ -ring map. Moreover, for k = 15 , the H ∞ -ring structure of Σ ∞ + KL is not induced byan E ∞ -ring spectrum. As the homotopy category of E ∞ -ring spectra is equivalent to the homotopy category ofcommutative ring spectra, for example by [15, Chapters II.3 and 4], this indeed gives rise to anexample of a G ∞ -ring spectrum whose structure is not induced from an ultra-commutative ringspectrum.Tyler Lawson provides a series of examples of H ∞ -ring spectra that do not come from commu-tative ring spectra in his paper [25], where he uses an algebraic reformulation of an H ∞ -structurein terms of Dyer-Lashof-operations, explicitly described in [25, Theorem 4]. Using this and theresults of [29] on E ∞ -ring spectra, he deduces the following theorem as [25, Theorem 1]: Theorem 3.34.
Let R k be a wedge of Eilenberg-Mac Lane spectra such that π ∗ R k is isomorphicto the graded ring F [ x ] / ( x ) , where | x | = − k . Then R k has an H ∞ -ring structure, and for k > , these structures are not induced from commutative ring spectra. For more details on these examples, the reader is referred to the cited articles.
In this section, we generalize the analysis of the symmetric powers P m X classically provided byLemma 3.17, comparing them to the extended powers ( E Σ m ) + ∧ Σ m X ∧ m . Thus, we connect our32efinition of G ∞ -ring spectra using the derived monad G to the original definition of H ∞ -ringspectra using the extended powers. As in the G -equivariant version of Lemma 3.17 given in[19, Proposition B.117], we need to replace the universal space E Σ m with an appropriate globalobject. The correct analogue is the global universal space E gl Σ m defined in Remark 2.1.Recall that the global universal space E gl Σ m is constructed as L V , where V is a faithful Σ m -representation. Then, its suspension spectrum can be described by the untwisting isomorphismΣ ∞ + L V → O ( V, _) ∧ S V = F V A (3.35)as defined in (2.2). This isomorphism Σ ∞ + L V → F V S V induces the isomorphism Σ ∞ + L G,V ∼ = F G,V S V from (2.3) on G -orbits.We also consider the morphism λ G,V,W : F G,V ⊕ W S V → F G,V (3.36)defined in [34, 4.1.28] for any compact Lie group G and G -representations V and W with W faithful. At an inner product space U , the map λ G,V,W is represented as O ( V ⊕ W, U ) ∧ G S V → O ( W, U ) /G [( u, ϕ ) , t ] [ u + ϕ ( t ) , ϕ ◦ i ] , where i : W → V ⊕ W is the inclusion as the second factor. The map λ G,V,W is a globalequivalence by [34, Theorem 4.1.29].
Theorem 3.37.
Let X be a positively flat orthogonal spectrum, and n ≥ . Then the map q = q Xn : Σ ∞ + L R n ∧ Σ n X ∧ n → X ∧ n / Σ n = P n X that collapses Σ ∞ + L R n to S = Σ ∞ + ∗ is a global equivalence.Proof. For the proof, we use a Σ n -equivariant decompositionΣ ∞ + L R n ∼ = F R n S n j ←− ( F R S ) ∧ n . The isomorphism j arises from the homeomorphism ( S ) ∧ n ∼ = S n and the isomorphism ( F R ) ∧ n ∼ = F R n from [34, Remark C.11]. This decomposition is Σ n -equivariant, since both of the involvedmaps are symmetric. Explicitly, this isomorphism is given at inner product spaces U , . . . , U n as( F R S )( U ) ∧ . . . ∧ ( F R S )( U n ) j U ,...,Un −−−−−−→ ( F R n S n )( U ⊕ . . . ⊕ U n )[( u , ϕ ) , t ] ∧ . . . ∧ [( u n , ϕ n ) , t n ] [( u ⊕ . . . ⊕ u n , ϕ ⊕ . . . ⊕ ϕ n ) , t ∧ . . . ∧ t n ] . Using this decomposition, we can rewrite the domain of the morphism q as follows:Σ ∞ + L R n ∧ Σ n X ∧ n ∼ = F R n S n ∧ Σ n X ∧ n ∼ =( F R S ) ∧ n ∧ Σ n X ∧ n ∼ =( F R S ∧ X ) ∧ n / Σ n We claim that under this translation, the morphism q corresponds to the morphism( λ Σ , R , ∧ X ) n / Σ n : ( F Σ , R S ∧ X ) n / Σ n → ( F Σ , ∧ X ) n / Σ n = e is the trivial group, and hence 0 is a faithful Σ -representation.Moreover, F Σ , = O (0 , _) / Σ ∼ = S is the sphere spectrum. To prove this claim, we consider thediagram ( F R S ) ∧ n ∧ Σ n X ∧ n ( F R S ∧ X ) ∧ n / Σ n F R n S n ∧ Σ n X ∧ n Σ ∞ + L R n ∧ Σ n X ∧ n X n / Σ n . j ∧ Σ n X ∧ n ( λ Σ1 , R , ∧ X ) ∧ n / Σ n untwisting λ Σ1 , R n, ∧ X ∧ n q (3.38)We first consider the upper diagram. Let U , . . . , U n be inner product spaces, and consider thediagram ( F R S )( U ) ∧ . . . ∧ ( F R S )( U n ) O (0 , U ) ∧ . . . ∧ O (0 , U n )( F R n S n )( U ⊕ . . . ⊕ U n ) O (0 , U ⊕ . . . ⊕ U n ) ∼ = S U ⊕ ... ⊕ U n λ R , ( U ) ∧ ... ∧ λ R , ( U n ) j U ,...,Un ∼ = ⊕ λ R n, ( U ⊕ ... ⊕ U n ) Evaluating this on an element yields[( u , ϕ ) , t ] ∧ . . . ∧ [( u n , ϕ n ) , t n ] [ u + ϕ ( t )] ∧ . . . ∧ [ u n + ϕ n ( t n )][( u ⊕ . . . ⊕ u n , ϕ ⊕ . . . ⊕ ϕ n ) , t ∧ . . . ∧ t n ] [( u ⊕ . . . ⊕ u n ) + ( ϕ ( t ) ⊕ . . . ⊕ ϕ n ( t n ))]By applying (_) ∧ Σ n X ∧ n to this diagram, we see that the upper half of (3.38) commutes. Forthe second half of the Diagram (3.38), let U be an inner product space. We need to consider theleft diagram O ( R n , U ) ∧ S n S U ∧ L ( R n , U ) O (0 , U ) = S Uλ Σ1 , R n, ( U )untwisting q ( U ) [( u, ϕ ) , t ][ u + ϕ ( t ) , ϕ ] [ u + ϕ ( t )] . On elements, this takes the right form.Thus, also this part of the Diagram (3.38) commutes. Hence, we have translated the statementof the theorem into the claim that the map P n ( λ Σ , R , ∧ X ) = ( λ Σ , R , ∧ X ) ∧ n / Σ n : P n ( F Σ , R S ∧ X ) → P n X is a global equivalence. But by [34, Theorem 4.1.29], the morphism λ Σ , R , is a global equivalence.As the spectrum X is positively flat, smashing with X preserves global equivalences by [34,Theorem 4.3.27]. Moreover, we know by Lemma 3.9 that P n sends global equivalences betweenpositively flat spectra to global equivalences. As both X and F Σ , R S = F R S are positively flat,this proves that P n ( λ Σ , R , ∧ X ) is a global equivalence, and hence also q is. Remark . In this proof, we used a decomposition of the global classifying space Σ ∞ + L R n andthe fact that we already know that P n preserves global equivalences by Lemma 3.9 to give an easyproof of the theorem. Our proof thus relies on the fact that we already have a model structure on34ommutative ring spectra, following the approach of White in [45] via analysing the symmetricpowers P n . More classically, in [15], [28] and [37], the theorems analogous to Theorem 3.37 areused to provide the model structure on commutative ring spectra. In these sources, the proofof the above theorem is done by a cellular induction, see for example the proof of [15, TheoremIII.5.1]. A similar proof can also be done in our context, using the above calculations for theinduction start and then using [17, Theorem 22] for the induction over the cell attachments.Via this alternative proof, which does not use the property that P n preserves global equiva-lences between positively flat spectra, we could also give a new proof that P n preserves globalequivalences by proving this for the extended symmetric powers Σ ∞ + L R n ∧ Σ n X n . Remark . The above theorem allows us to give an alternative description of G ∞ -ring spectra,which is closer to the original definition of [12]. In this version, a G ∞ -ring spectrum is a spectrum X together with structure maps ζ m : D gl m X = Σ ∞ + L R m ∧ L Σ m X ∧ L m → X in the global homotopycategory, such that compatibility diagrams similar to those of Proposition 1.3 commute in GH .The translation to the original definition of G ∞ -ring spectra works by precomposition with theglobal equivalence q . 35 Power operations on Moore spectra and β -rings In this part of the article, we study G ∞ -structures on global Moore spectra S B for torsion-freecommutative rings B . We show that G ∞ -ring structures on S B provide β -ring structures onall equivariant homotopy groups. The reason that we study Moore spectra is first of all thatthey are (almost) completely determined by the underlying algebra of the ring B , so we cantranslate the topological structure of being a G ∞ -ring spectrum into an algebraic structure on B . Moreover, a Moore spectrum can be thought of as an extension of coefficients of the spherespectrum and hence has relevance to talking about cohomology theories with coefficients in aring B . Hence, providing power operations in the Moore spectrum S B is a first step to providingpower operations on these extended cohomology theories.It is known classically that torsion in the ring B obstructs the existence of a highly structuredmultiplication on the Moore spectrum S B . We can show, however, that these obstructions alsooccur algebraically. Hence, we restrict our analysis to the class of torsion-free rings, where thesephenomena are not visible.We start this chapter with an algebraic analysis of power operations on global Green functorsthat are “induced from the trivial group”, or equivalently are scalar extensions of a global powerfunctor by non-equivariant data. This analysis provides a notion of power operations on a ring,indexed by a global power functor. This new notion is linked to the classical notion of λ - and β -rings, giving a new perspective on these objects. We exhibit this relation in Section 4.2.After having established this connection between β -rings and scalar extension of global powerfunctors, we consider in Section 4.3 the topological side of the situation and construct from aglobal power structure on the homotopy groups of a Moore spectrum a G ∞ -ring structure. InTheorem 4.43, we arrive at an equivalence between the homotopy category of Moore spectrafor torsion-free rings equipped with a G ∞ -structure and the category of corresponding ringsequipped with power operations indexed by the Burnside ring global power functor.A similar analysis has already been done by Julia Singer in her PhD-thesis [36, Chapter2.4] in the case of H ∞ -structures. There, the representation ring functor R takes the role ofthe Burnside ring. Our treatment is based on her ideas, generalizing the arguments to a globalcontext, and algebraically from the quite restrictive class of functors studied in the cited thesisto all global power functors.In this chapter, all rings B are commutative and unital. F ⊗ F ( e ) B We consider the following situation:
Definition 4.1.
Let B be a unital, associative and commutative ring, and let F be a globalGreen functor. Then we define the global functor F ⊗ B as the composite A F −→ Ab (_) ⊗ B −−−−→ Ab . Explicitly, on a compact Lie group this takes the value F ( G ) ⊗ B .If B is a F ( e )-algebra, we define F ⊗ F ( e ) B analogously. Here, we consider F ( G ) as an F ( e )-algebra via the × -product, which gives a map F ( G ) × F ( e ) → F ( G ).Note that for any compact Lie group G , F ( G ) ⊗ B is a unital, associative and commutativering, using the multiplications on F ( G ) and B . With this ring structure, the restrictions are ringmorphisms and the transfers satisfy Frobenius reciprocity, as this holds for F . All in all, we seethat F ⊗ B is a global Green functor. 36f B is a F ( e )-algebra, then F ⊗ F ( e ) B is a global Green functor as well. We are mostly interestedin the case F = A , and in this case, these two functors agree, since A ( e ) ∼ = Z . However, weformulate much of the algebraic part of this section with a general global functor F , since it isa straightforward generalization. In this case, we need to work with an F ( e )-algebra B and therelative tensor product F ⊗ F ( e ) B , in order to arrive at structure for B and not for F ( e ) ⊗ B byspecializing to the trivial group.The construction F ⊗ F ( e ) (_) is left adjoint to evaluating global functors at the trivial group.To formalize this, we introduce the relative notion of an algebra over a global Green or powerfunctor. Definition 4.2.
Let F be a global Green (or power) functor. Then an F -Green (power) algebrais a global Green (power) functor R together with a map η : F → R of global Green (power)functors. We denote the corresponding categories by G l G reen F and G lP ow F .If it is clear from the context whether we consider Green or power functors, we omit thisfrom the notation. The main example of an F -Green algebra is F ⊗ F ( e ) B , and the question weinvestigate is under which conditions on B this extends to an F -power algebra if F is a globalpower functor. Another example of an F -power algebra is exp( F ) defined in [34, Chapter 5.2],endowed with the power operations P : F → exp( F ) as a structure map. This map is compatiblewith the power operations by the coassociativity diagram for the coalgebra structure map P . Proposition 4.3.
Let F be a global Green functor. Then the functors Alg F ( e ) G l G reen FF ⊗ F ( e ) ( _ )ev e are adjoint, with the extension functor F ⊗ F ( e ) ( _ ) left and the evaluation ev e at the trivial groupright adjoint.The unit of this adjunction is the canonical isomorphism B ≃ −→ F ( e ) ⊗ F ( e ) B, b ⊗ b and the adjunction counit is the multiplication morphism α : F ⊗ F ( e ) R ( e ) η ⊗ F ( e ) R ( e ) −−−−−−−→ R ⊗ F ( e ) R ( e ) × −→ R for the F -Green algebra R .Proof. For the proof, we check that the two composites F ⊗ F ( e ) B F ⊗ ⊗ B −−−−−→ F ⊗ F ( e ) F ( e ) ⊗ F ( e ) B ×⊗ B −−−→ F ⊗ F ( e ) B and R ( e ) ⊗ R ( e ) −−−−−→ F ( e ) ⊗ F ( e ) R ( e ) × −→ R ( e )of unit and counit are the identity. But both these maps insert a 1 and then multiply, hencedescribe the identity.This adjunction takes place on the level of F -Green algebras and F ( e )-algebras. Our aim isto lift this adjunction to an adjunction where we replace the F -Green algebras by power algebras.We can describe power algebras by the comonad exp : G l G reen → G l G reen defined in [34, Chapter5.2], hence we consider whether we can transport this comonad along this adjunction. This is37one generally in Section A.2. There, we have to restrict to those objects where the unit orcounit of the adjunction is an isomorphism, and obtain a weak coalgebra structure. Note thatsince the counit of the adjunction is the multiplication map α : F ⊗ F ( e ) R ( e ) → R, we exactly restrict our attention to F -Green algebras of the form F ⊗ F ( e ) B , where B = R ( e ). Remark . We know that the category G lP ow is the category of coalgebras for the comonad expon G l G reen . A priori, it is not clear that the same is true for the F -relative versions G l G reen F and G lP ow F when F is a global power functor. However, if F η −→ R is an F -Green algebra, wecan endow exp( R ) with the structure of an F -Green algebra via F P F −−→ exp( F ) exp( η ) −−−−→ exp( R ).Then, we see that the structure of an F -power algebra is the same as a morphism R → exp( R )making R into an exp-coalgebra, since the diagram F R exp( F ) exp( R ) ηP F P R exp( η ) commutes. Thus, we can also describe G lP ow F as the category of coalgebras over exp.This argument can also be carried out formally, using a distributive law of the monad F (cid:3) (_),describing F -Green algebras, over the comonad exp, describing global power functors. The formalframework of distributive laws between comonads and monads is developed in [32, Chapter 6].In our case, the distributive law is the morphism λ : F (cid:3) exp( R ) P (cid:3) id −−−→ exp( F ) (cid:3) exp( R ) → exp( F (cid:3) R ).Applying the formalism from Section A.2 to this situation, we obtain the following: Definition 4.5.
Let F be a global power functor. An F -powered algebra is a commutative F ( e )-algebra B together with a map τ : B → exp( F ⊗ F ( e ) B ; e )of F ( e )-algebras, such that the diagrams B exp( F ⊗ F ( e ) B ; e ) B exp( F ⊗ F ( e ) B ; e )exp( F ⊗ F ( e ) exp( F ⊗ F ( e ) B ; e ); e ) B = F ( e ) ⊗ F ( e ) B exp( F ⊗ F ( e ) B ; e ) exp(exp( F ⊗ F ( e ) B ); e ) τ ε ( e ) ττ exp( F ⊗ τ ; e )exp( α exp( F ⊗ F ( e ) B ) ; e ) ν ( e ) (4.6)commute.A morphism of F -powered algebras is a morphism f : B → C of F ( e )-algebras such that B C exp( F ⊗ F ( e ) B ; e ) exp( F ⊗ F ( e ) C ; e ) fτ B τ C exp( F ⊗ f ; e ) commutes. We denote the category of F -powered algebras by P owAlg F .38 emark . We make some of the above structure a bit more transparent. We consider the “weakcomonad” B exp( F ⊗ F ( e ) B ; e ). The counit is the morphism exp( F ⊗ F ( e ) B ; e ) → F ( e ) ⊗ F ( e ) B ∼ = B that sends an exponential sequence x to its constant term x . The comultiplication ofthis weak comonad differs from the one of exp by the map exp( α exp( F ⊗ F ( e ) B ) ; e ). This is inducedby the map α exp( F ⊗ F ( e ) B ) : F ⊗ F ( e ) exp( F ⊗ F ( e ) B ; e ) P ⊗ id −−−→ exp( F ) ⊗ F ( e ) exp( F ⊗ F ( e ) B ; e ) × −→ exp( F ⊗ F ( e ) B ) . As a special case of Theorem A.16, we obtain the following characterization of global powerfunctors of the form F ⊗ F ( e ) B : Theorem 4.8.
Let F be a global power functor. Let G lP ow left F denote the full subcategory ofthose F -power algebras R such that the multiplication map F ⊗ F ( e ) R ( e ) → R is an isomorphism.Then the functors P owAlg F G lP ow left FF ⊗ F ( e ) ( _ )ev e are inverse equivalences of categories. Hence, we have described the exact structure we need on a ring B such that F ⊗ F ( e ) B isan F -power algebra. We can also explicitly describe the induced power operations on a globalpower functor F ⊗ F ( e ) B for an F -powered algebra B . Proposition 4.9.
Let F be a global power functor and B be an F -powered algebra with structuremap τ . Then the induced power operations of the global power functor F ⊗ F ( e ) B are given by P m ( x ⊗ b ) =∆ ∗ m ( P mF ( x ) × τ m ( b ))= P mF ( x ) · (Σ m ≀ p G ) ∗ τ m ( b ) for any element x ⊗ b ∈ F ( G ) ⊗ F ( e ) B . Here ∆ m : Σ m ≀ G → Σ m ≀ G × Σ m is the diagonal and p G : G → e is the morphism to the trivial group.Proof. Applying Proposition A.13, the power operations on F ⊗ F ( e ) B are described by the map F ⊗ F ( e ) B F ⊗ τ −−−→ F ⊗ F ( e ) exp( F ⊗ F ( e ) B ; e ) α exp( F ⊗ F ( e ) B ) −−−−−−−−−→ exp( F ⊗ F ( e ) B ) . Using the description of α in Remark 4.7, we see that an element x ⊗ b ∈ F ( G ) ⊗ F ( e ) B is sentto P F ( x ) × τ ( b ) ∈ exp( F ⊗ F ( e ) B ; G ) by the total power operation P . Restricting to the m -thcomponent yields ( P F ( x ) × τ ( b )) m = ∆ ∗ m ( P mF ( x ) × τ m ( b )) , using the definition of the multiplication on exp.The second description follows from a similar calculation.We now make the structure of an F -powered algebra more explicit, by considering the struc-ture morphism τ : B → exp( F ⊗ F ( e ) B ; e ) levelwise. We translate the property of τ making B into an F -powered algebra into properties similar to those of the power operations of a globalpower functor. Proposition 4.10.
Let F be a global power functor, B be an F ( e ) -algebra and τ = ( τ m ) m ≥ : B → Y m ≥ F (Σ m ) ⊗ F ( e ) B be any map. Then the following are equivalent: The map τ factors through exp( F ⊗ F ( e ) B ; e ) and defines the structure of an F -powered algebraon B . The maps τ m satisfy the following conditions:i) τ m (1) = 1 for the units ∈ B on the left hand side and ∈ F (Σ m ) ⊗ F ( e ) B on the righthand side.ii) τ = Id B as a map B → F (Σ ) ⊗ F ( e ) B ∼ = B , and τ ( b ) = 1 for all b ∈ B .iii) For all m ≥ , we have τ m ( a · b ) = τ m ( a ) · τ m ( b ) in F (Σ m ) ⊗ F ( e ) B .iv) For all x ∈ F ( e ) and b ∈ B and all m ≥ , we have τ m ( x × b ) = P mF ( x ) · τ m ( b ) in F (Σ m ) ⊗ F ( e ) B .v) For all i, j ≥ and b ∈ B , we have Φ ∗ i,j ( τ i + j ( b )) = τ i ( b ) × τ j ( b ) in F (Σ i × Σ j ) ⊗ F ( e ) B .vi) For all k, m ≥ and b ∈ B , we have Ψ ∗ k,m ( τ km ( b )) = ( P kF · ((Σ k ≀ p Σ m ) ∗ ◦ τ k ))( τ m ( b )) in F (Σ k ≀ Σ m ) ⊗ F ( e ) B . Here, the map P kF · ((Σ k ≀ p Σ m ) ∗ ◦ τ k ) : F (Σ m ) ⊗ B → F (Σ k ≀ Σ m ) ⊗ B sends an element x ⊗ b to P kF ( x ) · (Σ k ≀ p Σ m ) ∗ ( τ k ( b )) .vii) For all a, b ∈ B and m ≥ , we have τ m ( a + b ) = m X k =0 tr k,m − k ( τ k ( a ) × τ m − k ( b )) in F (Σ m ) ⊗ F ( e ) B .Proof. Note that this proposition is, up to point iv ) and vi ), completely analogous to [34, The-orem 5.2.13], and the proof is a straightforward translation of the properties: The fact that τ factors through exp( F ⊗ F ( e ) B ; e ) is equivalent to the statement that every ( τ m ( b )) m ≥ is anexponential sequence, which is point v ) and the assertion τ ( b ) = 1 from ii ). The map τ isadditive exactly if property vii ) is satisfied, and a map of (unital) F ( e )-algebras exactly if theproperties i ), iii ) and iv ) are satisfied.The map τ makes the counit diagram from (4.6) commute exactly when τ is the identity of B ,as the counit ε : exp → Id is the projection on the linear summand. As the comultiplication ν is defined using the maps Ψ ∗ k,m and the coassociativity diagram incorporates the map α , whichuses the power operations in the first variable and then multiplies, we see that the coassociativitydiagram from (4.6) is equivalent to property vi ). This finishes the proof. Remark . Using this proposition, we see that our notion of F -global powered algebra agreeswith the notion of a τ -ring with respect to F from [16, Definition 4.3], which itself is a general-ization of the notion from [20]. 40 xample 4.12. We give some examples of F -powered algebras and some examples which donot support power operations. i) For any F -power algebra R , evaluation of the power operations at the trivial group definesthe structure of an F -powered algebra on R ( e ), since ev e is part of the equivalence above.In particular, the integers Z support power operations both for the Burnside ring globalpower functor A and for the representation ring global functor. ii) In [16, Section 4.2], an F -powered algebra structure on F ( G ) is constructed for any globalpower functor. Note however, that this is done via the construction F ( G ) P −→ exp( F ; G ) ⊂ Y m ≥ F (Σ m ≀ G ) ( δ Gm ) ∗ −−−−→ Y m ≥ F (Σ m × G ) × ←− Y m ≥ F (Σ m ) ⊗ F ( e ) F ( G ) , where δ Gm : Σ m × G → Σ m ≀ G sends ( σ, g ) to ( σ ; g, . . . , g ). Then, it is used that in the contextof [16], the product map × is assumed to be an isomorphism. This is not the case in ourcontext, as for example the Burnside ring global functor does not have this property. Thus,working in the category of global power functors, it is not possible to define a canonical F -powered algebra structure on F ( G ) from the power operations on F .However, if we take F to be the representation ring global functor, then × is an isomorphism,and F ( G ) is an F -powered algebra. iii) We claim that the rings Z /n do not support A -power operations for any n . Suppose theydo support such an operation τ : Z /n → Y m ≥ A (Σ m ) ⊗ Z /n. Note that since Z /n is additively cyclic, the entire operation τ is determined on the unit1 ∈ Z /n by the additivity relation. Moreover, we have τ m (1) = 1, so there is at most one A -power operation on Z /n . Furthermore, if it exists, it is induced by the A -power operationson Z = A ( e ) upon taking the quotient by n , as this satisfies additivity. Hence, we canprovide formulas for the power operations in terms of powers of finite sets. We claim thatif p is any prime factor of n , then the power operation τ p : Z → A (Σ p ) does not descend to Z /n → A (Σ p ) ⊗ Z /n .To check this, we have to calculate P p ( n ) ∈ A (Σ p ) ∼ = Z { conjugacy classes of subgroups of Σ p } and check whether all coefficients are divisible by n . We observe by the explicit descriptionof the power operations on the Burnside ring of a finite group in [34, Example 5.3.1] thatthe element P p ( n ) ∈ A (Σ p ) is represented by the Σ p -set [ n ] p with the permutation action,where we denote by [ n ] the set { , . . . n } with no group actions. We identify the coefficientsin A (Σ p ) via decomposing a Σ p -set as a disjoint union of Σ p -orbits, which are of the formΣ p /H for subgroups H occurring as isotropy groups of elements.We calculate the coefficient of e ⊂ Σ p , which is the number of free Σ p orbits inside [ n ] p .These orbits are in bijection with equivalence classes of points ( k , . . . , k p ) with pairwisedifferent k i , up to reordering. There are (cid:16) np (cid:17) such classes, hence this is the coefficient ofΣ p /e in [ n ] p . Since we have (cid:18) np (cid:19) = n · . . . · ( n − p + 1) p · . . . · n is divisible by p , we see that (cid:16) np (cid:17) is divisible by np , but not by n . Thus the element P p ( n ) ∈ A (Σ p ) is not divisible by n . Hence we do not have A -poweroperations on the rings Z /n for any n .These rings do however support truncated power operations, namely Z /n supports maps P k for all k < p , where p is the smallest prime divisor of n . This can be shown by similarcomputations as above for all stabilizers of points in [ n ] p . Such truncated power operationsare also an object of research, for example in [7]. β -rings We now connect the theory of A -powered algebras to the theory of β -rings. We accomplish thisvia a pairing A (Σ m ≀ G ) ⊗ A (Σ m ) → A ( G ) on the Burnside rings of symmetric groups, allowing usto “dualize” the structure morphism C → exp( A ⊗ C ; e ). This pairing arises from the additionalstructure of deflations present in A , which are transfers for surjective group homomorphisms.Our approach is loosely based on the discussion of τ -rings in [16, Section 4.2], which itself goesback to [20].We start by describing the deflations for the Burnside ring global functor. These turn A into a global functor with deflations, which are a special case of Webb’s globally defined Mackeyfunctors, for the maximal families of restrictions and transfers [44, Chapter 8]. In fact, thediscussion in this section can be carried out for a general global power functor with deflations.However, the main examples of such objects are the Burnside and representation ring globalfunctors. As the structure arising from using the representation rings, namely λ -rings, is wellunderstood, we focus on the Burnside ring global functor. Remark . We give a short overview over the history of the notion of β -rings.The notion of a β -ring was first introduced by Rymer in [33], based on the question posed byBoorman in [9] whether there is a theory of β -rings which formalizes the β -operations on theBurnside ring defined in this work. Rymer, however, did not define his operator ring structureon B = L m ≥ A (Σ m ) properly, a fact explained and amended by Ochoa in [31]. An explicitconstruction of the β -operations, using the language of polynomial operations, is given by Vallejoin [42], and he extended the definition of a β -ring by a unitality condition in [43].Lastly, the survey article [18] of Guillot provides more details on the history of β -rings and theirconnection to λ -rings, as well as showing that cohomotopy is an example of a β -ring. Moreover,there an additivity condition is added to the notion of β -rings. Construction 4.14.
The deflations in the Burnside ring global functor are described as follows,in the case of a finite group: Let f : G → K be a group homomorphism, and X be a finite G -set.Then we define f ∗ ( X ) = K × G X = ( K × X ) /G , where we identify ( k · f ( g ) , x ) with ( k, gx ) forall g ∈ G , and consider f ∗ ( X ) as a K -set by the multiplication on the K -factor. This defines amap A ( G ) → A ( K ). For the general case of a morphism G → K of compact Lie groups, thisconstruction is generalized in [40, Proposition 20] and [41, Proposition IV.2.18]. The interactionswith the remaining structure of the global functor A is exhibited in [11] for finite groups, andpackaged in [44, Chapter 8] in the notion of a globally defined Mackey functor. Definition 4.15.
We denote B = M m ≥ A (Σ m ) . We endow this abelian group with a commutative multiplication via x · y = tr Σ k + l Σ k × Σ l ( x × y )42or x ∈ A (Σ k ) and y ∈ A (Σ l ). Moreover, we define an operation ∗ on B as follows: Let x ∈ A (Σ k )and y i ∈ A (Σ l i ) for 1 ≤ i ≤ n . Then we define x ∗ ( y + . . . + y n ) = X ( k ) tr Σ ( k ) · ( l ) × Σ ki ≀ Σ li n × i =1 P k i ( y i ) · (( × Σ k i ≀ p Σ li ) ∗ Φ ∗ ( k ) )( x ) ! , (4.16)where the sum runs over all partitions of k , we denote ( k ) · ( l ) := P ni =1 k i l i and the transfer isalong the monomorphisms Ψ k i ,l i and Φ ( k i l i ) i . This map is additive in the first component andcan be extended linearly to give a map ∗ : B × B → B .The operation ∗ is sometimes called plethysm, for example in [31] and in [20] for the repre-sentation ring instead of the Burnside ring. It makes B into an operator ring by [43, Theorem1.11]. We then define a β -ring as an “operator module” over B . The following definition is givenin [43, Definition 1.12]: Definition 4.17. A β -ring is a commutative ring A together with a map ϑ : B → Map(
A, A )such that, for all a ∈ A and x, y ∈ B , the following relations hold: i) ϑ ( x + y ) = ϑ ( x ) + ϑ ( y ) ii) ϑ ( x · y ) = ϑ ( x ) · ϑ ( y ) iii) ϑ ( x ∗ y ) = ϑ ( x ) ◦ ϑ ( y ) iv) ϑ (1)( a ) = 1 for the multiplicative unit 1 ∈ A (Σ ) ⊂ B on the left and 1 ∈ A on the right v) ϑ ( e ) = id A , where e = 1 ∈ A (Σ ) ⊂ B is a unit for the operation ∗ .In the first two statements, we use the pointwise ring structure on Map( A, A ).For the construction of a β -ring structure from an A -powered structure on A , we need thefollowing pairing on the Burnside ring. Construction 4.18.
For compact Lie groups K and G , we define a pairing h _ , _ i K,G : A ( K × G ) ⊗ A ( K ) → A ( G )as the composition A ( K × G ) ⊗ A ( K ) × −→ A ( K × G × K ) ∆ ∗ K −−→ A ( K × G ) (pr G ) ∗ −−−−→ A ( G ) . Here ∆ K : K × G → K × G × K is the diagonal of K . Note that this pairing can also be describedas A ( K × G ) ⊗ A ( K ) id ⊗ pr ∗ K −−−−−→ A ( K × G ) ⊗ A ( K × G ) · −→ A ( K × G ) (pr G ) ∗ −−−−→ A ( G )by the translation between internal and external product via projections and diagonal.Since we are ultimately interested in a pairing A (Σ m ≀ G ) ⊗ A (Σ m ) → A ( G ), we use the morphism δ Gm : Σ m × G → Σ m ≀ G, ( σ, g ) ( σ ; g, . . . , g )to define A (Σ m ≀ G ) ⊗ A (Σ m ) ( δ Gm ) ∗ ⊗ id −−−−−−→ A (Σ m × G ) ⊗ A (Σ m ) h _ , _ i Σ m,G −−−−−−−→ A ( G )43he pairing h _ , _ i K,G is biadditive and multiplicative. Moreover, it is natural for restrictionsin the G -variable and satisfies Frobenius-reciprocity in the K -variable.Using these definitions, we can define for any commutative ring C a morphism D G : exp( A ⊗ C ; G ) → Map( B , A ( G ) ⊗ C ) x = ( x n ) n ≥ ( y n ) X n ≥ h ( δ Gn ) ∗ x n , y n i . (4.19)Here, the pairing h _ , _ i only uses the A -part of x n and is the identity on C . Proposition 4.20.
Let C be a commutative ring and G be a compact Lie group. Then for any x ∈ exp( A ⊗ C ; G ) , the morphism D G ( x ) is a ring homomorphism.Proof. Additivity of the morphism D G ( x ) : B → A ( G ) ⊗ C is clear from biadditivity of the pairing h _ , _ i . Moreover, let 1 ∈ A (Σ ) be the multiplicative unit of B . Then we calculate D G ( x )(1) = h ( δ G ) ∗ x , i = p ∗ G ( x ) = 1 , using that exponential sequences have as zeroth term the unit of A ( e ) ⊗ C and that δ G is theunique map p G : G → e .Now, for two elements y, z ∈ B , we have D G ( x )( y · z ) = X n ≥ D ( δ Gn ) ∗ x n , X k + l = n tr Σ k + l Σ k × Σ l ( y k × z l ) E = X k,l ≥ h res Σ k + l × G Σ k × Σ l × G ( δ Gk + l ) ∗ x k + l , y k × z l i = X k,l ≥ h ∆ ∗ G (( δ Gk ) ∗ x k × ( δ Gl ) ∗ x l ) , y k × z l i = X k,l ≥ h ( δ Gk ) ∗ x k , y k i · h ( δ Gl ) ∗ x l , z l i = D G ( x )( y ) · D G ( x )( z ) . Here, we use that the morphisms δ Gk + l ◦ (Φ k,l × G ) and Φ Gk,l ◦ ( δ Gk × δ Gl ) ◦ ∆ G : Σ k × Σ l × G → Σ k + l ≀ G agree and that the sequence x is exponential.Using this morphism, we can now study how any A -powered algebra C induces the structureof a β -ring on A ( G ) ⊗ C : Construction 4.21.
Let C be an A -powered algebra with structure map τ : C → exp( A ⊗ C ; e ).Let G be a compact Lie group. Then we define¯ ϑ G : A ( G ) ⊗ C P −→ exp( A ⊗ C ; G ) D G −−→ Hom
Rings ( B , A ( G ) ⊗ C ) , where P denotes the power operation induced on A ⊗ C from τ .This map is adjoint to a map ϑ : B → Map( A ( G ) ⊗ C, A ( G ) ⊗ C ) . Proposition 4.22.
The map ϑ makes A ( G ) ⊗ C into a β -ring. roof. It is already known that A ( G ) is a β -ring by [33, Theorem 2], where the relations areonly checked for some additive generators of B , and by [43, Corollary 1.16], where this resultis extended to the entirety of B using the theory of polynomial operations. It is clear from thedefinition that the map ϑ agrees with the one defined by Rymer, hence induces the structureof a β -ring on A ( G ). Moreover, the condition that C comes equipped with the structure of an A -powered algebra extends these calculations to A ( G ) ⊗ C . Thus, ϑ endows A ( G ) ⊗ C with thestructure of a β -ring. Remark . Note that the result that A ( G ) is a β -ring is not new, as already the above proofshows. However, the above definition of the structure map θ illustrates how this structure can beobtained from a more naturally arising structure, namely from the global power functor structureon A . We hope that this allows for a clearer picture of β -rings.The definition of a β -ring only incorporates relations between the different operations ϑ ( x )on A ( G ) ⊗ C , indexed by x ∈ B . It does not prove any compatibility of these operations withthe ring structure on C . We now add a condition of “external” additivity, due to [18]: Definition 4.24.
We define B = M p,q ≥ A (Σ p × Σ q ) . This has a ring structure analogous to the one on B , given by x · y = tr Σ p + r × Σ q + s Σ p × Σ q × Σ r × Σ s ( x × y ) for x ∈ A (Σ p × Σ q ) and y ∈ A (Σ r × Σ s ). Moreover, we have mapsΦ : B → B , x X p + q = m Φ ∗ p,q x for x ∈ A (Σ m )and × : B ⊗ B → B , x ⊗ y x × y for x ∈ A (Σ p ) , y ∈ A (Σ q ) . Additivity is then expressed by a morphism ϑ : B → Map( A × A, A ) analogous to ϑ , usingthe codiagonal Φ. Definition 4.25.
An additive β -ring is a commutative ring A with maps ϑ : B → Map(
A, A )and ϑ : B → Map( A × A, A ) such that (
A, ϑ ) is a β -ring and the following properties hold: i) ϑ ( x × y )( c, d ) = ϑ ( x )( c ) · ϑ ( y )( d ) for all x, y ∈ B and all c, d ∈ A . ii) ϑ ( x )( c + d ) = ϑ (Φ x )( c, d ) for all x ∈ B and c, d ∈ A .We construct such a map ϑ for A ( G ) ⊗ C with C an A -powered algebra. Construction 4.26.
Let C be an A -powered algebra. Then we define D : exp( A ⊗ C ; G ) × exp( A ⊗ C ; G ) → Hom( B , A ( G ) ⊗ C )( x, y ) z X p,q ≥ h ( δ Gp,q ) ∗ ( x p × y q ) , z p + q i , where δ Gp,q = ( δ Gp × δ Gq ) ◦ ∆ G : Σ p × Σ q × G → Σ p ≀ G × Σ q ≀ G is the diagonal on G .Moreover, we define¯ ϑ : ( A ( G ) ⊗ C ) × ( A ( G ) ⊗ C ) P × P −−−→ exp( A ⊗ C ; G ) × exp( A ⊗ C ; G ) D −−→ Hom( B , A ( G ) ⊗ C ) , and denote the morphism adjoint to ¯ ϑ as ϑ : B → Map(( A ( G ) ⊗ C ) × ( A ( G ) ⊗ C ) , A ( G ) ⊗ C ) . roposition 4.27. The morphisms ϑ and ϑ makes A ( G ) ⊗ C into an additive β -ring.Proof. We only prove part ii ) from Definition 4.25, since the first assertion is an easy calculation,using the description δ Gp,q = ( δ Gp × δ Gq ) ◦ ∆ G .We thus calculate ϑ ( x )( c + d ) = X m ≥ h ( δ Gm ) ∗ P m ( c + d ) , x m i = X m ≥ D ( δ Gm ) ∗ X p + q = m tr p,q ( P p ( c ) × P q ( d )) , x m E = X p,q ≥ h tr Σ p + q × G Σ p × Σ q × G ( δ Gp,q ) ∗ ( P p ( c ) × P q ( d )) , x p + q i = X p,q ≥ h ( δ Gp,q ) ∗ ( P p ( c ) × P q ( d )) , Φ ∗ p,q x p + q i = D ( P ( c ) × P ( d ))(Φ x ) = ϑ (Φ x )( c, d ) . Here, in the third line, we use the observation that there is only one double coset inΣ p + q × G \ Σ p + q ≀ G/ Σ p ≀ G × Σ q ≀ G, and hence the double coset formula for ( δ Gp + q ) ∗ tr p,q consists of a single summand.Finally, we consider in which sense this construction is functorial. We can study functorialityboth in the A -powered algebra C and in the compact Lie group G . Definition 4.28.
Let A and A ′ be additive β -rings with structure morphisms ϑ, ϑ and ϑ ′ , ϑ ′ .Then a morphism f : A → A ′ of β -rings is a ring homomorphism f such that the relations f ( ϑ ( x )( a )) = ϑ ′ ( x )( f ( a )) and f ( ϑ ( y )( a , a )) = ϑ ′ ( y )( f ( a ) , f ( a ))hold for all x ∈ B , y ∈ B and a, a , a ∈ A . Proposition 4.29. i) Let G be a compact Lie group and f : B → C be a morphism of A -powered algebras. Then A ( G ) ⊗ f is a morphism between the β -rings A ( G ) ⊗ B and A ( G ) ⊗ C .ii) Let ϕ : K → G be a homomorphism of compact Lie groups and C be an A -powered algebra.Then ϕ ∗ ⊗ C is a morphism between the β -rings A ( G ) ⊗ C and A ( K ) ⊗ C .Proof. For the first assertion, we calculate for x ∈ B and b ∈ A ( G ) ⊗ B : ϑ C ( x )(( A ( G ) ⊗ f )( b )) = D G ( P C (( A ( G ) ⊗ f )( b )))( x ) = X n ≥ h ( δ Gn ) ∗ P nC (( A ( G ) ⊗ f )( b )) , x n i = X n ≥ h ( A ( G ) ⊗ f )( δ Gn ) ∗ P nB ( b ) , x n i = X n ≥ ( A ( G ) ⊗ f ) h ( δ Gn ) ∗ P nB ( b ) , x n i = ( A ( G ) ⊗ f )( ϑ B ( x )( b )) . Here, in the last line, we used that the pairing h _ , _ i only uses the A -component.For the second assertion, we suppress the ⊗ C from the notation. Then we calculate for x ∈ B and c ∈ A ( G ) ⊗ C : ϑ K ( x )( ϕ ∗ ( c )) = X n ≥ h ( δ Kn ) ∗ P n ( ϕ ∗ ( c )) , x n i X n ≥ h ( δ Kn ) ∗ (Σ n ≀ ϕ ) ∗ P n ( c ) , x n i = X n ≥ h ( G × ϕ ) ∗ ( δ Gn ) ∗ P n ( c ) , x n i = X n ≥ ϕ ∗ h ( δ Gn ) ∗ P n ( c ) , x n i = ϕ ∗ ϑ G ( x )( c ) . The corresponding calculations with ϑ work analogously.Thus, we have proven the following result, where we denote by Rep the category of com-pact Lie groups and conjugacy classes of continuous group homomorphisms and by β -Rings thecategory of additive β -rings. Theorem 4.30.
The assignment ( G, C ) A ( G ) ⊗ C extends to a functor A ( _ ) ⊗ _ : Rep op × P owAlg A → β -Rings , which sends a conjugacy class of a morphism of compact Lie groups to the corresponding restric-tion. In this theorem, we only treat restrictions. In fact, transfers do not induce morphisms of β -rings. The reason is that transfers do not commute with the morphism ( δ Gn ) ∗ .We give one example of a β -ring which come up in the literature. Example 4.31.
We apply our theory to the A -powered algebra C = Z = A ( e ). Then we obtain β -ring structures on the Burnside rings A ( G ) for all compact Lie groups G . The operations hereare given as follows:For the element x = Σ n /H ∈ B = L n ≥ A (Σ n ), we obtain for finite G and a finite G -set X theformula β H ( X ) := β Σ n /H ( X ) = h P n ( X ) , Σ n /H i = Σ n /H × Σ n X n = X n /H, where we consider the resulting set as a G -set. This formula agrees with the one from [42] andgeneralizes to compact Lie groups as shown in [33]. Thus, in this case, we obtain the classical β -ring structure on A ( G ), using our abstract definition. Also the iterated operations β used foradditivity agree with those defined in [18, Example 3.2]. In fact, we have β H = ( X p × Y q ) /H for a finite group G , finite G -sets X and Y and H ⊂ Σ p × Σ q .The above construction of β -rings already highlights the importance of the notion of an A -powered algebra. Theorem 4.30 shows that the notion of an A -powered algebra encodes com-patible β -ring structures for all compact Lie groups at once. The comparison 4.30 is not perfect,however. It remains open to what extent we can represent all β -rings by A -powered algebras,for example. In general, the condition of having an A -powered structure on a ring is strongerthan admitting a β -ring structure. Also, we require no multiplicative behaviour of the opera-tions ϑ . In face of the complications posed in the analysis of β -rings, starting with finding afeasible definition, it seems sensible to propose that the notion of an A -powered ring is the morefundamental one, since it arises naturally from considering power operations on the ring C froma global viewpoint. 47 .3 G ∞ -structure on Moore spectra for powered algebras In this section, we use the algebraic results from Section 4.1 to construct G ∞ -ring structureson Moore spectra. It is well known that the existence of a multiplication on Moore spectra isa subtle question, which traces back to the fact that the cone of a map is only well definedin the homotopy category up to non-canonical isomorphism. This distinguishes them notablyfrom Eilenberg-MacLane spectra, which are unique in a more rigorous way and hence are better-behaved for algebraic manipulations. However, there are classes of rings which work better thanothers, in particular, torsion-free rings support multiplications on their Moore spectra. We thusrestrict our attention to the subcategory of torsion-free rings in the following chapter. The mainexamples of Moore spectra for torsion rings that do not support commutative multiplicationsare S ( Z / S ( Z / S ( Z /n ) can have an A n -ring structure. That no Moore spectrum S ( Z /n ) canbe endowed with a G ∞ -structure is clear by our Example 4.12, where we show that Z /n doesnot admit the structure of a A -powered algebra. The relevant connection between A -poweredalgebras B and G ∞ -structures on S B is established in Corollary 4.34. For countable torsion-freerings B , Theorem 4.43 provides a full description of G ∞ -ring structures on S B via A -poweredstructures on B .In this chapter, we use the triangulated structure of the categories SH and GH . As a refer-ence for these triangulated structures we use [34, Chapter 4.4].Recall from [34, Theorem 4.5.1] that the forgetful functor U : GH → SH has both adjoints,and that the left adjoint is the left derived functor of the identity functor Id : S p → S p . Wecall the objects in the image of the left adjoint left induced homotopy types , and a spectrum leftinduced (from the trivial group) if its homotopy type is. When we talk about Moore spectra forrings, we require that all their homology is not only concentrated in degree 0, but also determinedby the non-equivariant ring. Hence we make the following definition: Definition 4.32.
Let B be a ring. A (global) Moore spectrum for B is a connective spectrum X , left induced from the trivial group, such that H e ( X ) ∼ = π e ( X ) ∼ = B and H e ∗ ( X ) = 0 for all ∗ 6 = 0. We denote a Moore spectrum for the ring B by S B .Since we study power operations on the homotopy groups of a Moore spectrum, we need tocalculate π ( S B ) as a global functor. This can be done in terms of B , using that S B is connectiveand left induced. Proposition 4.33.
Let X be a connective spectrum left induced from the trivial group. Thenthe exterior product ⊠ : π G ( S ) ⊗ π e ( X ) → π G × e ( S ∧ X ) ∼ = π G ( X ) is an isomorphism of abelian groups. As the group G varies, these assemble into an isomorphism π ( S ) ⊗ π e ( X ) → π ( X ) of global functors. If X is a homotopy ring spectrum, then this is an isomorphism of global Greenfunctors.Proof. We first observe that the proposition is true in the case X = S , since in this case π e ( S ) ∼ = Z ,and the exterior product is the multiplication map π G ( S ) ⊗ Z → π G ( S ) , X for which the exteriorproduct map is an isomorphism is closed under coproducts, since ⊠ is additive in the spectrum X . It is also closed under cones as defined before [34, Proposition 4.4.13], by the followingargument:Suppose we have a distinguished triangle X → Y → Z → X [1]of connective spectra in the stable homotopy category, where for X and Y , the map ⊠ : π G ( S ) ⊗ π e (_) → π G (_)is an isomorphism. Then the left-induced triangle is also distinguished, and we obtain a longexact sequence in homotopy groups π G ( X ) → π G ( Y ) → π G ( Z ) → π G − ( X ) = 0for all compact Lie groups G . As π G ( S ) is free by [34, Proposition 4.1.11] and the exteriorproduct is natural, we obtain the commutative diagram π G ( S ) ⊗ π e ( X ) π G ( S ) ⊗ π e ( Y ) π G ( S ) ⊗ π e ( Z ) 0 π G ( X ) π G ( Y ) π G ( Z ) 0 ⊠ ⊠ ⊠ with exact rows, where the two left vertical maps are isomorphisms. Then by the 5-lemma, alsofor Z the exterior product is an isomorphism.Then by [34, Proposition 4.4.13], respectively its non-equivariant analogue, the map ⊠ is anisomorphism for all connective left-induced spectra, since the sphere spectrum is a compactweak generator of SH .Now, by the properties of the external product [34, Theorem 4.1.22], the map ⊠ is a morphismof global functors, and levelwise an isomorphism, hence it is an isomorphism of global functors.Moreover, if X is a homotopy ring spectrum, then ⊠ is a map of global Green functors, sincethe product × : π G ( X ) × π K ( X ) → π G × K ( X ) is defined in terms of the exterior product ⊠ , andthis is compatible with the involved twist-morphisms. This proves the proposition.Using this proposition, we see that the homotopy-group global functor for a left-inducedspectrum is π ( X ) = A ⊗ π e ( X ). For functors of this form, we have the results of Section 4.1,so we immediately obtain the following result: Corollary 4.34.
Let X be a left induced spectrum. If X supports the structure of a G ∞ -ringspectrum, then it induces on π e ( X ) the structure of an A -powered algebra. In particular, π G ( X ) has the structure of a β -ring.Proof. We already know from Construction 3.6 that π ( X ) for a G ∞ -ring spectrum X supportspower operations, and Theorem 4.8 shows that evaluating a global power functor A ⊗ π e ( X )at the trivial group gives π e ( X ) the structure of an A -powered algebra. Then, Theorem 4.30provides a β -ring structure on π G ( X ) ∼ = A ( G ) ⊗ π e ( X ).We now study the reverse direction in the case that X is a global Moore spectrum. ByTheorem 4.8, the structure of a global power functor on A ⊗ B is equivalent to the structure of49n A -powered algebra on B . Now, we define a G ∞ -ring structure on S B from power operationson its homotopy groups. As already mentioned, we restrict to torsion-free rings in the following.We first show that the ring structure of B induces the structure of a homotopy ring spectrumon S B . For this, we show that we can test properties of morphisms between Moore spectra onthe homotopy groups: Lemma 4.35.
The functor π e : Moore torsion-free → Ab torsion-free between the homotopy category of Moore spectra of torsion-free abelian groups and the categoryof torsion-free abelian groups is fully-faithful, and hence an equivalence of categories.Proof. Let A and B be torsion-free groups. We have to calculate the group of morphisms[ S A, S B ] := GH ( S A, S B ). To do so, we consider a free resolution0 → Z ⊕ I → Z ⊕ J → A → A . Then we take as a model of the Moore spectrum for A the cone _ I S → _ J S → S A. Using this distinguished triangle and mapping into S B , we obtain an exact sequence "_ J Σ S , S B → "_ I Σ S , S B → [ S A, S B ] → "_ J S , S B → "_ I S , S B . Since the homotopy classes are additive under wedges, we can write the above sequence asHom( Z ⊕ J , π ( S B )) → Hom( Z ⊕ I , π ( S B )) → [ S A, S B ] → Hom( Z ⊕ J , B ) → Hom( Z ⊕ I , B ) . As Hom is left exact and the sequence (4.36) is exact, we see that the kernel of the rightmost mapis Hom(
A, B ). Moreover, the cokernel of the leftmost map is isomorphic to Ext Z ( A, π ( S B )).We now calculate π ( S B ), using a free resolution0 → Z ⊕ I ′ → Z ⊕ J ′ → B → , which gives a cofibre sequence in the global homotopy category. The associated long exactsequence in homotopy groups gives M I ′ Z / → M J ′ Z / → π ( S B ) → M I ′ Z → M J ′ Z , using π ( S ) ∼ = Z /
2. As the rightmost map is injective and tensor product is right-exact, weobtain π ( S B ) ∼ = B ⊗ Z / Z is additive, we see that Ext Z ( A, B/
2) is 2-torsion. Moreover, the sequence 0 → A −→ A → A/ → A is torsion-free, and the long exact sequence of Ext-functorsyields that Ext Z ( A, B/ −→ Ext Z ( A, B/ → Ext Z ( A/ , B/
2) = 0is exact. Thus Ext Z ( A, B/
2) is also 2-divisible and hence vanishes.50o consider ring structures on Moore spectra, we need to calculate whether S B ∧ S B is againa Moore spectrum. This follows from the following: Proposition 4.37.
Let A and B be commutative rings whose underlying abelian group is torsion-free. Then the external product H ∗ ( S A, Z ) ⊗ H ∗ ( S B, Z ) → H ∗ ( S A ∧ S B, Z ) in homology is an isomorphism. Hence S A ∧ S B is a Moore-spectrum for A ⊗ B .Proof. This follows from the Künneth-theorem, see for example [1, KT1 and Note 12]. Thehomology groups H ∗ ( S A ) and H ∗ ( S B ) are flat over Z , since they are torsion-free. This provesthat H ∗ ( S A ∧ S B ) is A ⊗ B concentrated in degree 0. Corollary 4.38.
The equivalence from Lemma 4.35 induces an equivalence π e : Moore torsion-freeRings → Rings torsion-free between the homotopy category of commutative homotopy ring Moore spectra for torsion-free ringsand the category of torsion-free commutative rings. In particular, there is a unique structure (upto equivalence) of a commutative homotopy ring spectrum on S B inducing the multiplication of B on π e for any torsion-free commutative ring B . We now follow the same approach in order to put a G ∞ -ring structure on S B . Note that G ( S B ) is not a Moore-spectrum, since it is not left induced. However, we still can calculate thegroup [ G ( S B ) , S B ] in terms of the homotopy groups of S B . Lemma 4.39.
Let B be a countable torsion-free abelian group, and let Y be an orthogonalspectrum such that for every finite group G , π G ( Y ) is finite. Then for any of the spectra X = G ( S B ) or X = G ( G ( S B )) , the morphism π : GH ( X, Y ) → GF ( π ( X ) , π ( Y )) (4.40) is an isomorphism.Proof. We start by considering G ( S B ), and first consider the case that B is free. Then we choosea basis ( x i ) i ∈ I of B , such that S B ∼ = W i ∈ I S . We calculate G m ( S B ) = G m _ i ∈ I S h x i i ! = _ ( m ) ^ i ∈ Imi =0 G m i S = _ ( m ) ^ i ∈ Imi =0 Σ ∞ + B gl Σ m i = _ ( m ) Σ ∞ + B gl × i ∈ Imi =0 Σ m i , (4.41)where ( m ) = ( m i ) i ∈ I runs through all partitions of m = P i ∈ I, m i =0 m i .By representability of the homotopy groups π G by Σ ∞ + B gl G from (2.4), the map π (4.40) is anisomorphism for the spectra Σ ∞ + B gl × i ∈ Imi =0 Σ m i .
51s both domain and codomain of the map (4.40) are additive under wedges, also for G ( S B ) themorphism π is an isomorphism.Let now B be a general torsion-free abelian group. By [14, Theorem A6.6], any flat moduleis isomorphic to a filtered colimit of finitely generated free modules, generated on finite sets ofelements in B . Since over Z , flat and torsion-free are equivalent, we can write B ∼ = colim i ∈ I B i as a filtered colimit of finitely generated free Z -modules. Moreover, since B is countable, we finda cofinal sequential system in the filtered indexing system, so that we can write B as a sequentialcolimit of finitely generated free modules B n with n ∈ N .We can then lift this sequential system to a cofibrant system S B n of Moore spectra, such thatthe colimit colim n S B n models the homotopy colimit. Since homology commutes with sequentialhomotopy colimits, we see that S B ∼ = colim n S B n is a model for the Moore spectrum of B .Since sequential colimits of ultra-commutative ring spectra are calculated on the underlyingspectra, also the functor P : S p → S p commutes with sequential colimits, and thus G ( S B ) ∼ =colim n G ( S B n ). Then, we obtain the Milnor exact sequence0 → lim n GH ( G ( S B n ) , Ω Y ) → GH ( G ( S B ) , Y ) → lim n GH ( G ( S B n ) , Y ) → . By the above arguments for free modules B , we see that the right hand object is isomorphic tolim n GH ( G ( S B n ) , Y ) ∼ = lim n Hom GF ( π ( G ( S B n )) , π ( Y )) ∼ = Hom GF ( π ( G ( S B )) , π ( Y )) . Similarly, the left hand object is isomorphic tolim n GH ( G ( S B n ) , Ω Y ) ∼ = lim n Hom GF ( π ( G ( S B n )) , π ( Y )) . By the calculations (4.41), this last term is isomorphic tolim n Y ( m n ) π Σ ( mn ) ( Y ) , where the product is over all tupels of natural numbers m ni ≥ I n of B n , andwe denote Σ ( m n ) = × i ∈ I n Σ m ni . Now this lim -term decomposes as the product Y m ≥ lim n YP In m ni = m π Σ ( mn ) ( Y ) . In each of the individual lim -terms, the inverse system consists only of finite groups, since I n is finite for every n and π G ( Y ) is finite for any finite group G . Thus, these systems satisfy theMittag-Leffler condition and thus the lim -term vanishes.This proves that the morphism π is an isomorphism.The arguments for G ( G ( S B )) are completely analogous.We now check that the assumption that π G ( Y ) is finite is satisfied in the case of a Moorespectrum S B . Lemma 4.42.
Let B be an abelian group and S B be a Moore spectrum for B . Theni) There is an isomorphism π ( S B ) ∼ = π ( S ) ⊗ B . i) For any compact Lie group G , we have π G ( S ) ∼ = M H ⊂ G Z / ⊕ ( π ( W G H )) ab , where the sum runs over all conjugacy classes of closed subgroups of G , W G H denotes theWeyl group of H in G and ( _ ) ab denotes abelianization.iii) For any finite group G , the homotopy group π G ( S ) is finite.Proof. Let 0 → Z ⊕ I → Z ⊕ J → B → B . Then, we construct the Moorespectrum S B as the mapping cone in _ I S → _ J S → S B. Thus, the long exact sequence of homotopy groups becomes . . . → M I π ( S ) → M J π ( S ) → π ( S B ) →→ M I π ( S ) → M J π ( S ) → π ( S B ) . Now, by freeness of π G ( S ) ∼ = A ( G ), we know that the second row of this sequence is left exact,so the first row is right exact. Moreover, tensoring with π ( S ) is right exact, hence applying thisto the sequence 0 → Z ⊕ I → Z ⊕ J → B → π ( S B ) ∼ = π ( S ) ⊗ B .For the second part, we use the tom Dieck splitting [40, Satz 2], which decomposes for anycompact Lie group G the homotopy groups as π G ( S ) ∼ = M ( H ) ⊂ G π W G H ( EW G H + ∧ S H ) . Here, the sum runs over conjugacy classes of closed subgroups of G , and W G H denotes the Weylgroup of H in G . Then, the Adams isomorphism [2, Theorem 5.4] identifies the right hand sideas M ( H ) ⊂ G π (Σ ∞ + BW G H ) . Now the based suspension spectrum of BW G H splits stably as S ∨ e Σ ∞ + BW G H , a sum of thesphere spectrum and the reduced suspension spectrum of BW G H . Thus, we find π (Σ ∞ + BW G H ) ∼ = π ( S ) ⊕ π st1 ( BW G H ) ∼ = Z / ⊕ π ( W G H ) ab . This is the required splitting of the first equivariant homotopy groups of the sphere spectrum.The third assertion is then a direct consequence of ii) . Theorem 4.43.
The functor π e : G ∞ -Moore torsion-free → P owAlg torsion-free A is an equivalence of categories between the homotopy category of G ∞ -Moore spectra for countabletorsion-free commutative rings and the category of countable torsion-free A -powered algebras. roof. We first prove that the functor π e from the theorem is essentially surjective. Thus, let B be a torsion-free A -powered algebra. We have to define a G ∞ -multiplication ζ : G ( S B ) → S B on S B .By Lemma 4.39, we know that the map π : GH ( G ( S B ) , S B ) → GF ( π ( G ( S B )) , π ( S B ))is an isomorphism. We have calculated the homotopy groups global functor of S B in Proposition4.33 to be A ⊗ B . Moreover, by [34, Theorem 5.4.11], we have that π ( G ( S B )) is the free globalpower functor on the global functor π ( S B ) ∼ = A ⊗ B . We denote this free global power functorby F ( A ⊗ B ).This free global power functor is part of a diagram G lP ow G l G reen GF U U UC P F of adjunctions, where all functors labelled U are forgetful functors and right adjoints, P is thesymmetric algebra functor for the box product of global functors and C is the free global powerfunctor for a global Green functor constructed in [34, Proposition 5.2.21]. We now claim thatthe composite adjunction featuring F is monadic. For this, we use Beck’s monadicity theorem,see for example [27, VI.7 Theorem 1].Let R ⇒ S be a pair of parallel arrows in G lP ow that has a split coequalizer in GF . Thenby monadicity of the adjunction G l G reen GF U P (see Lemma 1.2), the coequalizer in GF hasthe unique structure of a global Green functor such that it is a coequalizer of R ⇒ S in G l G reen .Since G lP ow is also comonadic over G l G reen by [34, Theorem 5.2.13], colimits in G lP ow arecreated by U : G lP ow → G l G reen . Thus, Beck’s monadicity theorem shows that the adjunction G lP ow GF UF is monadic. We denote the associated monad U F also by F .Hence, the power functor structure on A ⊗ B arising from the structure of an A -poweredalgebra on B is equivalent to a morphism τ : F ( A ⊗ B ) → A ⊗ B , satisfying the compatibilityconditions with the monad structure on F . Since π (4.40) is an isomorphism for G ( S B ) and S B , this morphism τ : F ( A ⊗ B ) → A ⊗ B is the image of a unique morphism ζ : G ( S B ) → S B under the functor π . We claim that ζ endows S B with the structure of a G ∞ -ring spectrum.For this, we check that the functor π : GH ≥ → GF sends the monad diagrams for G tothose of F . Here, the subscript ≥ GH ≥ Ho(ucom) ≥ GF G lP ow L gl P π π F and Ho(ucom) ≥ GH ≥ G lP ow GF . Uπ π U (4.44)The right diagram commutes, thus we get a natural transformation ρ : F ◦ π → π ◦ L gl P in theleft diagram as the mate of the right isomorphism (see [22, Proposition 2.1] for the definition ofmates). This transformation is thus defined by freeness of F , and [34, Theorem 5.4.11] provesthat ρ is a natural isomorphism. The pasting of the two diagrams in (4.44), using the inverse of54he transformation ρ , then exhibits π : GH ≥ → GF as a monad functor between the monads G and F . For the compatibility with the monad structure, naturality of mates is used.Thus, we see that the G ∞ -diagrams G ( G ( S B )) G ( S B ) G ( S B ) S B G ( ζ ) µ ζζ and S B G ( S B ) S B η ζ are sent under π to the corresponding diagrams for the monad F . Since τ defines the structureof an F -algebra on A ⊗ B by assumption, the monad diagrams for A ⊗ B commute. UsingLemma 4.39 for morphisms out of G ( G ( S B )) and Lemma 4.35 for S B , also the G ∞ -diagrams for S B commute. This proves that π e : G ∞ -Moore torsion-free → P owAlg torsion-free A is essentially surjective.To check that π e is also fully faithful, we only need to check that the unique induced map S B → S C from Lemma 4.35 for a map f : B → C of A -powered algebras is a morphism of G ∞ -ring spectra. But this again follows from Lemma 4.39 by looking at the diagrams G ( S B ) S B G ( S C ) S C ζ B ζ C F ( A ⊗ B ) A ⊗ BF ( A ⊗ C ) A ⊗ C τ B F f fτ C . Here, the right diagram commutes by assumptions on f , so also the left diagram commutes. Intotal, the functor π e : G ∞ -Moore torsion-free → P owAlg torsion-free A is an equivalence of categories. 55 Transfer results for monads and comonads
In this appendix, we collect results from the theory of monads and comonads we use throughoutthis paper.We use a derived symmetric algebra monad to define G ∞ -ring spectra in Definition 3.3. Anal-ogously, the non-equivariant H ∞ -ring spectra are defined as algebras over the derived symmetricalgebra monad in the stable homotopy category. To facilitate the analysis of these structures,we consider how monads and monad morphisms behave under taking derived functors in SectionA.1. This is done in the generality of 2- and double categories. We use the results from thissection to construct an adjunction between G ∞ - and H ∞ -ring spectra in Section 3.2.Secondly, we consider whether comonads can be transferred along adjunctions in Section A.2.This is utilized to analyse scalar extensions of global power functors in Section 4.1. A.1 Transferring monads under lax functors
In Section 3.2, we study two lifting theorems for functors between algebras over the derivedsymmetric algebra monad in the global and stable homotopy categories. To separate the homo-topy theoretic properties needed to provide the liftings from the formal background in monadtheory, it is convenient to use the language of 2-categories. We also recall aspects of doublecategories, which we use when we encounter both left and right derived functors. For the theoryof 2-categories, we refer to [22], [39] and [10, Chapter 7], for the theory of double categories, werefer to [22] and [35].
Definition A.1.
A 2-category is a category enriched in the category Cat of categories, and adouble category is a category object in Cat.Thus, explicitly, a 2-category consists of classes of objects, morphisms and transformations,where we have a horizontal composition ⋆ and a vertical composition ◦ of transformations,and compositions of morphisms is strictly unital and associative. For horizontal and verticalcomposition, we use the conventions( η : g → g ′ ) ⋆ ( ϑ : f → f ′ ) = X Y Z ff ′ ϑ gg ′ η and ( η : g → h ) ◦ ( ϑ : f → g ) = X Y. fg h ϑη
A double category consists of a class of objects, classes of horizontal and vertical morphisms eachbeing part of a category with common objects, and a class of transformations
X YZ W, ⇒ also called squares or 2-cells. Transformations can be composed both horizontally and vertically,and all possible orders of composition agree. We denote horizontal composition by ⊟ and verticalcomposition by ⊟ . Note that for any double category C , we obtain two 2-categories V ( C ) and H ( C ) by considering only vertical morphisms and 2-cells with identities as horizontal morphisms,or considering horizontal morphisms respectively.For these notions of higher categories, there exist various versions of functors and naturaltransformations between them. We need the following:56 efinition A.2. Let C and D be 2-categories. A lax 2-functor F : C → D consists of the followingdata: i) assignments X F ( X ), ( f : X → Y ) ( F ( f ) : F ( X ) → F ( Y )) and ( η : f → g ) ( F ( η ) : F ( f ) → F ( g )) of objects, morphisms and transformations, ii) and transformations α X : id F ( X ) → F ( id X ) and µ g,f : F ( g ) ◦ F ( f ) → F ( gf ) for any object X and any pair ( g, f ) of composable morphisms in C .These have to satisfy the compatibility conditions given in [10, Definition 7.5.1]. Definition A.3.
Let
F, G : C → D be two lax functors between 2-categories. A lax naturaltransformation η : F → G between F and G consists of assignments X ( η X : F ( X ) → G ( X )) and ( f : X → Y ) F ( X ) G ( X ) F ( Y ) G ( Y ) η X F f Gf ⇒ η f η Y , such that the compatibility conditions given in [10, Definition 7.5.2] are satisfied.For two lax transformations η : F → G and θ : G → H between lax 2-functors, the compositeis given by sending X to the morphism θ X ◦ η X and a morphism f : X → Y to the transformation F ( X ) G ( X ) H ( X ) F ( Y ) G ( Y ) H ( Y ) . η X F f Gf θ X ⇒ η f Hf ⇒ θ f η Y θ Y We now relate these notions to the theory of monads. First note that the definition of a monadcan be given in any 2-category, generalizing an endofunctor T : C → C to an endomorphism T : X → X of an object X and the multiplication and unitality natural transformations tocorresponding transformations. In the same way, the definition of a (lax) monad functor can begeneralized to the context of 2-categories. Applying this generalized definition to the 2-categoryof categories, we obtain again the classical notion of monads. The following observation goesback to [6, 5.4.1]: Lemma A.4.
Let C be a -category, and let be the terminal -category with a single object ∗ , itsidentity morphism and the identity natural transformation. Then, the category of lax -functors → C and lax natural transformations and the category of monads and lax monad morphisms in C are isomorphic via the functor ( T : 1 → C ) ( T ( id ∗ ) : T ( ∗ ) → T ( ∗ ) , µ : T ( id ∗ ) ◦ T ( id ∗ ) → T ( id ∗ ) , η : id T ( ∗ ) → T ( id ∗ ))( ρ : S → T ) ρ ∗ : S ( ∗ ) → T ( ∗ ) , ρ ( id ∗ ) : T ( ∗ ) T ( ∗ ) S ( ∗ ) T ( ∗ ) ρ ∗ S ( id ∗ ) T ( id ∗ ) ⇒ ρ ∗ . The proof is an easy translation of the corresponding properties.Using this description, we see that monads are preserved under any lax 2-functor, and laxmonad morphisms are preserved if we moreover assume that some of the structure maps of a laxfunctor are invertible. 57 orollary A.5.
Let C and D be -categories and let F : C → D be a lax -functor. Let ( T : X → X, ν : T ◦ T → T, ε : id X → T ) be a monad in C . Then ( F ( T ) : F ( X ) → F ( X ) , F ( ν ) ◦ µ T,T : F T ◦ F T → F T, F ( ε ) ◦ η X : id F X → F T ) is a monad in D .Moreover, let S, T be two monads in C on objects X and Y respectively, let ( f : X → Y, ρ : T f → f S ) be a lax monad morphism, and assume that the transformation µ f,S is invertible. Then ( F f : F X → F Y, µ − f,S ◦ F ( ρ ) ◦ µ T,f : F T ◦ F f → F f ◦ F S ) is a lax monad morphism between F S and
F T .Proof.
The first part of this corollary follows directly from the above lemma: We can considerthe monad T in C as a lax 2-functor T : 1 → C . Then, the composition F ◦ T : 1 → D is a lax2-functor, with coherence morphisms the composites of the coherence morphisms of F and T .Thus, F T is a monad in D , and the structure of the monad is exactly given by the describedmorphism.For the second part, we note that the above lemma shows that a lax monad morphism from S to T is the same as a lax natural transformation between the corresponding lax 2-functors 1 → C .It is an easy argument that a lax 2-functor with invertible transformation µ f,S preserves such atransformation.In a double category C , we use the same formalism to consider whether a correspondingnotion of weak double functor preserves monads and morphisms between them. We thus firstdefine the appropriate notion of a weak double functor. Definition A.6.
Let C and D be double categories. A lax-oplax double functor F : C → D consists of assignments of objects, horizontal 1-cells, vertical 1-cells and 2-cells of D to those of C , and the following coherence data: i) Invertible unitality 2-cells
F X F XF X F X F ( id X ) ⇒ α hX id F ( X ) and F X F XF X F X F ( id X ) id F ( X ) ⇒ α vX for any object X of C . ii) A 2-cell
F X F ZF X F Y F Z F ( gf ) ⇒ µ hg,f F f F g for any composable pair X f −→ Y g −→ Z of horizontal morphisms, and a 2-cell F X F XF YF Z F Z F ( gf ) F f ⇒ µ vg,f F g X f −→ Y g −→ Z of vertical morphisms.These coherence cells need to satisfy the unitality, associativity and naturality relations as writtendown in [35, Definition 6.1 v) and vi) ]. Remark
A.7 . Note that in [35], the direction of the vertical structure 2-cells is reversed. In thatwork, all of the above are assumed to be isomorphisms, so the direction of the cells is irrelevant.In our application, the 2-cells µ vg,f are not invertible in general, so we have to take care of theorientation.On the other hand, the unitality 2-cells α hX and α vX are assumed to be invertible. This allows usto obtain from a lax-oplax double functor a lax 2-functor V ( F ) : V ( C ) → V ( D ) by applying F to vertical morphisms, and by defining V ( F ) X XY Y f g ⇒ η = F X F XF X F XF Y F YF Y F Y. ⇒ ( α hX ) − F ( id X ) F f F g ⇒ F ηF ( id Y ) ⇒ α hX In the same way, we obtain an oplax 2-functor H ( F ) : H ( C ) → H ( D ).In fact, the constraint that all α hX are invertible can be used to strictify F into a lax-oplaxfunctor where α hX = id X holds. The main result is the following: Lemma A.8.
Let C and D be double categories and F : C → D be a lax-oplax double functor.Let X XY Y f g ⇒ θ and X X ′ Y Y ′ hg g ′ ⇒ ηk be -cells in C . Then, the -cells F X F X ′ F Y F Y ′ F hF f F g ′ ⇒ F ( θ ⊟ η ) F k and
F X F X F X ′ F X F XF Y F YF Y F Y F Y ′ F h ⇒ ( α hX ) − F g ′ ⇒ F ( η ) F ( id X ) F f F g ⇒ F ( θ ) F ( id Y ) ⇒ α hY F k in D agree. The analogous statement holds for F ( η ⊟ θ ) . roof. We use the following chain of pasting diagrams:
F X F X F X ′ F X F XF Y F YF Y F Y F Y ′ F h ⇒ ( α hX ) − F g ′ ⇒ F ( η ) F ( id X ) F f F g ⇒ F ( θ ) F ( id Y ) ⇒ α hY F k = F X F X F X ′ F X F X F X ′ F Y F Y F Y ′ F Y F Y F Y ′ F h ⇒ ( α hX ) − ⇒ id Fh F ( id X ) F f F g F h ⇒ F ( θ ) F g ′ ⇒ F ( η ) F ( id Y ) F k ⇒ α hY ⇒ id Fk F k = F X F X ′ F X F X F X ′ F Y F Y F Y ′ F Y F Y ′ F h ⇒ µ hh,id F ( id X ) F f F g F h ⇒ F ( θ ) F g ′ ⇒ F ( η ) F ( id Y ) F k ⇒ ( µ hk,id ) − F k = F X F X ′ F Y F Y ′ . F hF f F g ′ ⇒ F ( θ ⊟ η ) F k
Here, we used the unitality and naturality conditions on a lax-oplax double functor in the secondand third step respectively. Also note that the unitality condition guarantees that the transfor-mation µ hk,id is indeed invertible.Now, we define the relevant notions of monads and morphisms between them in a double cat-egory. In our application, we have a left derivable monad and a right derivable monad morphism,and this motivates the following definition. Moreover, we also define monadic transformationsbetween monad morphisms in this context. Definition A.9.
Let C be a double category. A vertical monad T in C is a monad in thevertical 2-category V ( C ). A horizontal monad morphism between two vertical monads S and T on objects X and Y respectively is a horizontal morphism F : X → Y together with a 2-cell X YX Y,
FS T ⇒ ρF satisfying the unitality and mulitplicativity conditions X Y YX Y Y
FS T ⇒ ρ ⇒ η T F = X X YX X Y
S F ⇒ η S ⇒ id F F X X YX YX X Y
S S F ⇒ µ S T ⇒ ρFS T ⇒ ρF = X Y YYX Y Y.
FS T ⇒ ρ T ⇒ µ T TF For two horizontal monad morphisms (
F, ρ ) and (
G, σ ) between S and T , a monadic transfor-mation is a 2-cell X YX Y F ⇒ ηG such that the exchange relation X YX YX Y
FS T ⇒ ρF ⇒ ηG = X YX YX Y F ⇒ ηGS T ⇒ σG holds. Proposition A.10.
Let C and D be two double categories and let L : C → D be a lax-oplaxdouble functor. Let ( S, µ, η ) be a vertical monad in C . Then ( V ( L )( S ) , V ( L )( µ ) ◦ µ S,S , V ( L )( η ) ◦ α vX ) is a vertical monad in D .Moreover, let S and T be vertical monads in C and let ( F, ρ ) be a horizontal monad morphismbetween them. Then ( LF, Lρ ) is a horizontal monad morphism between V ( L )( S ) and V ( L )( T ) .Furthermore, for any monadic transformation η : F → G between two monad morphisms, thenatural transformation H ( L )( η ) is a monadic transformation between LF and LG .Proof. The first part is a direct consequence of (A.5), since a vertical monad is a monad in thevertical 2-category V ( C ) and L induces a lax 2-functor V ( L ) : V ( C ) → V ( D ).We now prove the second part. We first check the unitality condition on ( LF, Lρ ), and thusconsider
LX LY LY LYLY LYLY LYLX LY LY LY
LFLS ⇒ Lρ ⇒ α − Y ⇒ α Y L ( id Y ) LT L ( id Y ) ⇒ L ( η T ) L ( id Y ) ⇒ α Y LF =61 LX LY LYLX LY LY
LFLS L ( id Y ) ⇒ L ( ρ ⊟ η T ) ⇒ α Y LF = LX LY LYLX LY LY
LFLS L ( id Y ) ⇒ L ( η S ⊟ id F ) ⇒ α Y LF = LX LX LY LYLX LXLX LXLX LX LY LY LF ⇒ α − X L ( id Y ) ⇒ L ( id F ) ⇒ α Y LFLS L ( id X ) ⇒ L ( η S ) LF ⇒ α X LF = LX LX LX LYLX LXLX LXLX LX LX LY. ⇒ α − X LFL ( id Y ) ⇒ α X ⇒ id LF LFLS L ( id X ) ⇒ L ( η S ) LF ⇒ α X LF Here, we use Lemma A.8 in the first and third step.For the multiplicativity condition, we consider
LX LX LX LYLX LX LX LYLX LXLX LX LX LY ⇒ α − X LS LF ⇒ µ S,S LT ⇒ L ( ρ ) L ( id X ) LS L ( SS ) ⇒ L ( µ S ) LFLS LT ⇒ L ( ρ ) L ( id X ) ⇒ α X LF = LX LX LY LYLX LX LYLX LXLX LX LY LY LF ⇒ α − X L ( T T ) ⇒ L ( ρ ⊟ ρ ) LT ⇒ µ T,T L ( id X ) LS L ( SS ) ⇒ L ( µ S ) LTL ( id X ) ⇒ α X LF = LX LY LYLYLX LY LY
LFLS LT ⇒ L ( µ S ⊟ ( ρ ⊟ ρ ) | {z } ρ ⊟ µT ) LT ⇒ µ T,T
LTLF = LX LY LY LYLY LY LYLY LYLX LY LY LY.
LFLS ⇒ L ( ρ ) ⇒ α − Y LT ⇒ µ T,T L ( id Y ) LT L ( T T ) ⇒ L ( µ T ) LTL ( id Y ) ⇒ α Y LF This proves that (
LF, L ( ρ )) is a horizontal monad morphism.The fact that H ( L )( η ) is a monadic transformation between LF and LG is proven in the sameway, using a horizontal version of (A.8) for the exchange relation.Using this proposition, we also consider how the structure transformations of a lax-oplaxdouble functor behave for a composite of monad functors. Note that the composition of monad62unctors ( F, ρ ) : R → S and ( G, σ ) : S → T is a monad functor GF, X Y ZX Y Z
FR GS ⇒ ρ T ⇒ σF G . Lemma A.11.
Let C and D be double categories, R : X → X, S : Y → Y and T : Z → Z bevertical monads in C and let ( F, ρ ) be a horizontal monad morphism from R to S and ( G, σ ) be a horizontal monad morphism from S to T . Let moreover L : C → D be a lax-oplax doublefunctor. Then the structure map µ hG,F : L ( GF ) → LG ◦ LF is a monadic transformation in D between monad morphisms from V ( L )( R ) to V ( L )( T ) .Moreover, the unitality transformation α hX : L ( id X ) → id LX is a monadic transformation betweenmonad endomorphisms of V ( L )( R ) Proof.
We have to check that the equations
LX LZLX LY LZLX LY LZ L ( GF ) ⇒ µ hG,F LFLR LGLS ⇒ Lρ LT ⇒ LσLF LG = LX LZLX LZLX LY LZ L ( GF ) LR LT ⇒ L ( ρ ⊟ σ ) L ( GF ) ⇒ µ hG,F LF LG and
LX LXLX LXLX LX L ( id X ) ⇒ α hX LR LR ⇒ id LR = LX LXLX LXLX LX L ( id X ) LR LR ⇒ L ( id R ) L ( id X ) ⇒ α hX of 2-cells in D holds. But those are part of the naturality constrains of a lax-oplax double functor,see [35, 6.2]. A.2 Transferring a comonad along an adjunction
We now have considered under which circumstances we can transfer a monad along a functor.In Section 4.1, we also consider a comonad, namely the comonad exp describing global powerfunctors in the category of global Green functors. We relate this to an algebraic structure in thecategory of rings, using an adjunction between rings and Green functors. We study the relevanttheory in a general setup.Suppose we have an adjunction
C D LU with unit η : Id → U L and counit α : LU → Id. Let moreover (
G, ε, ν ) be a comonad on D with counit ε : G → Id and comultiplication ν : G → G . Using the structure maps of G and the63djunction, we define zig-zags in place of actual structure maps for a comonad on U GL : C → C ,leading to a “weak comonad”. We consider the zig-zags
U GL
UεL −−−→
U L η ←− Idand
U GL
UνL −−−→
U GGL
UGαGL ←−−−−−
U GLU GL.
Definition A.12.
Let (
L, U, η, α ) be an adjunction between C and D and let ( G, ε, ν ) be acomonad on D . Then a weak U GL -coalgebra is an object A of C together with a map τ : A → U GL ( A ), such that the counitality and coassociativity diagrams A U GL ( A ) U L ( A ) τη A Uε LA and A U GL ( A ) U GL ( A ) U GL ( U GL ( A )) U GGL ( A ) ττ Uν LA UGL ( τ ) UGα GL ( A ) commute.A morphism of weak U GL -coalgebras is a morphism f : A → B such that A U GL ( A ) B U GL ( B ) τ A f UGL ( f ) τ B commutes. We denote the category of weak U GL -coalgebras by coAlg
UGL .This notion is strongly related to that of G -coalgebras, as the adjunction C D LU translatesa weak U GL -coalgebra into a G -coalgebra, and under some conditions we can reverse this. Westart by proving that L preserves coalgebras. Proposition A.13.
Let ( L, U, η, α ) be an adjunction between C and D and let ( G, ε, ν ) be acomonad on D .Let ( A, τ ) be a weak U GL -coalgebra. Then the map LA Lτ −−→ L ( U GL ( A )) α GL ( A ) −−−−−→ GL ( A ) adjoint to τ endows LA with the structure of a G -coalgebra.Moreover, if f : A → B is a map of weak U GL -coalgebras, then Lf is a map between the induced G -coalgebras.Proof. We need to check the corresponding counitality and coassociativity diagrams. The couni-tality diagram takes the form
LA L ( U GL ( A )) G ( LA ) LU L ( A ) LA.
LτLη A α GL ( A ) LU ( ε LA ) ε LA α LA τ , and the right half commutes bynaturality of α . The composition along the lower left edge is the identity by the triangle equalityfor adjunctions.The coassociativity diagram is LA L ( U GL ( A )) GL ( A ) L ( U GL ( A )) L ( U GL ( U GL ( A ))) L ( U GGL ( A )) GL ( A ) GL ( U GL ( A )) GGL ( A ) . LτLτ α GL ( A ) L ( Uν LA ) ν LA L ( UGL ( τ )) α GL ( A ) LUGα GL ( A ) α GL ( UGL ( A )) α GG ( A ) GL ( τ ) Gα GL ( A ) In this diagram, the upper left pentagon commutes by the coassociativity diagram for the weak
U GL -coalgebra A . The other three squares commute by naturality of the adjunction counit α .Thus, LA together with the adjoint of τ has the structure of a G -coalgebra.To see that Lf is a morphism of G -coalgebras, we observe that the diagram LA L ( U GL ( A )) GL ( A ) LB L ( U GL ( B )) GL ( B ) Lτ A Lf α GL ( A ) L ( UGL ( f )) GL ( f ) Lτ B α GL ( B ) commutes by naturality and since f is a morphism of weak U GL -coalgebras.For a reverse statement, we need to restrict attention to objects on which the unit or thecounit of the adjunction is an isomorphism.
Definition A.14.
Let (
L, U, η, α ) be an adjunction between C and D . We call an object X of D left induced if the counit α X : LU X → X is an isomorphism. We call an object A of C rightinduced if the unit η A is an isomorphism. We denote by D left , coAlg left G , C right and coAlg right UGL thecorresponding full subcategories on the left respectively right induced objects, where (
G, ε, ν ) isa comonad on D .By the triangle equalities, the functors L and U restrict to give an equivalence C right D left . LU Moreover, combined with the above calculation this proves that L also lifts to a functor˜ L : coAlg right UGL → coAlg left G . Proposition A.15.
Let ( L, U, η, α ) be an adjunction between C and D and let ( G, ε, ν ) be acomonad on D . Let ( X, σ : X → GX ) be a G -coalgebra such that α X : LU X → X is an iso-morphism. Then ( U X, U X Uσ −−→ U GX
UGα − X −−−−−→ U GL ( U X )) is a weak U GL -coalgebra. Moreover,any G -coalgebra morphism f : X → Y between left induced G -coalgebras induces a morphism U f : U X → U Y of weak
U GL -coalgebras. roof. We again check the counitality and coassociativity diagrams. The counitality diagramtakes the form
U X U G ( X ) U GL ( U X ) U X U GL ( U X ) . Uση UX UGα − X Uε X Uε LU ( X ) Uα − X In this diagram, the upper left triangle commutes by counitality of σ , and the upper right squarecommutes by naturality. Moreover, the two lower diagonal morphisms agree, since ηU is a rightinverse of U α by the triangle equality for adjunctions.For the coassociativity, we consider
U X U G ( X ) U GL ( U X ) U G ( X ) U G ( X ) U GG ( X ) U GL ( U X ) U GL ( U GX ) U GGL ( U X ) .U GL ( U GL ( U X )) UσUσ UGα − X Uν X Uν LU ( X ) UGα − X UGσ UGGα − X UGL ( Uσ ) UGα X UGα GX UGL ( UGα − X ) UGα GL ( UX ) In this diagram, the upper left square commutes by coassociativity of σ , and the other squarescommute by naturality of ν and α . Thus, U X together with the specified structure map is aweak
U GL -coalgebra.For a G -coalgebra map f : X → Y between left induced G -coalgebras, the commutativity of thediagram U X U G ( X ) U GL ( U X ) U Y U G ( Y ) U GL ( U Y ) Uσ X Uf UGα − X UGf UGL ( Uf ) Uσ Y UGα − Y proves that U f is a morphism of weak
U GL -coalgebras.Thus, we also have a lift ˜ U : coAlg left G → coAlg right UGL . This is in fact an inverse to the lift ˜ L of L . Theorem A.16.
Let ( L, U, η, α ) be an adjunction between C and D and let ( G, ε, ν ) be a comonadon D . Then the two functors coAlg right UGL coAlg left G ˜ L ˜ U are inverse equivalences of categories, with the adjunction unit and counit inducing natural iso-morphisms ˜ η : Id → ˜ U ˜ L and ˜ α : ˜ L ˜ U → Id . roof. By definition of left and right induced objects, it is clear that η and α restrict to naturalisomorphisms on the underlying objects and morphisms. Thus we only need to check that thesetransformations are compatible with the structure morphisms. For this, we consider the diagrams A U GL ( A ) U L ( A ) U L ( U GL ( A )) U GL ( A ) U GL ( U L ( A )) τη A η UGL ( A ) UGL ( η A ) UL ( τ ) Uα GL ( A ) UGα − LA for the unit and LU ( X ) LU ( GX ) L ( U GL ( U X )) GL ( U X ) X GX LU ( σ ) α X LU ( Gα − X ) α GX α GL ( UX ) Gα X σ for the counit. Both of these diagrams commute by naturality and the triangle equalities foradjunctions. 67 eferences [1] J. F. Adams. Lectures on generalised cohomology. In Category Theory, Homology Theoryand their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three) ,pages 1–138. Springer, Berlin, 1969.[2] J. F. Adams. Prerequisites (on equivariant stable homotopy) for Carlsson’s lecture. In
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