Multiplicative equivariant K-theory and the Barratt-Priddy-Quillen theorem
Bertrand J. Guillou, J. Peter May, Mona Merling, Angélica M. Osorno
aa r X i v : . [ m a t h . A T ] F e b MULTIPLICATIVE EQUIVARIANT K -THEORY AND THEBARRATT-PRIDDY-QUILLEN THEOREM BERTRAND J. GUILLOU, J. PETER MAY, MONA MERLING,AND ANG´ELICA M. OSORNO
Abstract.
We prove a multiplicative version of the equivariant Barratt-Priddy-Quillen theorem, starting from the additive version proven in [13]. The proofuses a multiplicative elaboration of an additive equivariant infinite loop spacemachine that manufactures orthogonal G -spectra from symmetric monoidal G -categories. The new machine produces highly structured associative ringand module G -spectra from appropriate multiplicative input. It relies on newoperadic multicategories that are of considerable independent interest and aredefined in a general, not necessarily equivariant or topological, context. Mostof our work is focused on constructing and comparing them. We construct amultifunctor from the multicategory of symmetric monoidal G -categories tothe multicategory of orthogonal G -spectra. With this machinery in place, weprove that the equivariant BPQ theorem can be lifted to a multiplicative equiv-alence. That is the heart of what is needed for the presheaf reconstruction ofthe category of G -spectra in [12]. Contents
1. Introduction 21.1. A road map 51.2. Acknowledgements 72. Preliminaries on operads and multicategories 82.1. V -categories 82.2. Based V -categories 82.3. Operads in Cat ( V ) 102.4. Review of multicategories 113. The multicategory of O -algebras 123.1. The intrinsic pairing of an operad 123.2. Pseudo-commutative operads 143.3. The multicategory of O -algebras 153.4. Variants of Mult ( O ) and comparisons 183.5. The free O -algebra multifunctor O + Mathematics Subject Classification.
Primary 19D23, 19L47, 55P48; Secondary 18D20,18D40, 18M65, 55P91, 55U40.
Key words and phrases. K -theory, multiplicative equivariant infinite loop spaces, operads,multicategories, multifunctors.B. J. Guillou was partially supported by Simons Collaboration Grant No. 282316 and NSFgrants DMS-1710379 and DMS-2003204. M. Merling was partially supported by NSF grantDMS-1709461/1850644, a Simons AMS travel grant, and NSF CAREER grant DMS-1943925.A. M. Osorno was partially supported by the Simons Collaboration Grant No. 359449, theWoodrow Wilson Career Enhancement Fellowship, and NSF grant DMS-1709302. V ∗ -2 categories and their algebras and pseudoalgebras 224.1. V ∗ -2-categories 224.2. Algebras and pseudoalgebras over V ∗ -2-categories 254.3. Permutative structures on V ∗ -2-categories 295. The multicategory of D -algebras and the multifunctor R F F D -algebras 355.4. Definition of the functor R R is a symmetric multifunctor 406. The multicategory of D G -algebras and the multifunctor P G V -categories and G V ∗ -categories 416.2. Categories of operators over F G F G F G D G -algebras 476.6. The symmetric multifunctor P D G -algebras to F G -pseudoalgebras 507.1. E ∞ G -operads 517.2. The section map ζ G from F G to D G ζ ∗ G F G -algebras in Cat to G -spectra 619.1. The multifunctor B F G - G -spaces to G -spectra 639.3. The identification of objects in O - Alg G T ps O -transformations to homotopies of maps of G -spectra 6910. The multiplicative Barratt-Priddy-Quillen Theorem 7010.1. The construction of α α is a stable equivalence 7110.3. The proof that α is monoidal 7411. Coherence axioms 7511.1. Coherence axioms for pseudo-commutative operads 7511.2. Coherence axioms for Mult ( O ) 7811.3. Coherence axioms for Mult ( D ) 8212. The pseudo-commutativity of D ( O ) 83References 891. Introduction
We can view algebraic K -theory as a machine that takes as input a category witha structured additive operation and produces a spectrum by group-completing the ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 3 operation in a homotopy coherent way. The homotopy groups of this spectrum—thehigher K -groups—are rich invariants which connect homotopy theory with numbertheory, algebraic geometry, and geometric topology. For example, the homotopygroups of the K -theory spectrum of the category of finitely generated projective R -modules for a ring R are Quillen’s higher K -groups of R , which are relatedto important problems and conjectures in number theory, especially when R is anumber ring.Classically, there were two approaches for building the K -theory spectrum asso-ciated to a symmetric monoidal category: Segal’s approach based on Γ-spaces [47],and the operadic approach of [2, 29, 30]. These two infinite loop space machineswere shown to be equivalent in [37, 31]. One fundamental problem in infinite loopspace theory is to determine what structure on the input category ensures that its K -theory spectrum is a highly structured ring spectrum. If the input has a sec-ond, related, structured multiplicative operation, making it into a “ring category”,then a suitably multiplicative K -theory machine should yield a ring spectrum. Thestudy of multiplicative infinite loop space theory saw much development early on[32, 33, 38, 50, 51]. A space level modernized survey is given in [35] and a mod-ernized categorical treatment is given in [34]. A treatment of multiplicative infiniteloop space theory that is structured around the use of multicategories and multi-functors is given in [8], and that has served for inspiration in this paper.For a finite group G , the Segal infinite loop space machine has been generalizedequivariantly by Shimakawa in [48], and the operadic infinite loop space machinehas been generalized equivariantly by two of us in [13] to build (genuine) orthogonal G -spectra from categories with additive operations that are suitably equivariant. These equivariant infinite loop space machines have been shown to be equivalentby three of us in [39].It is a natural question to ask what kind of structure on a G -category makesits K -theory into an equivariant ring spectrum, and this is not addressed in any ofthe papers just mentioned. Nonequivariantly, the question can be answered withoutserious use of 2-category theory, but we have not found such an answer equivariantly.The multiplicative structure at the categorical level is encoded via multilinear mapsthat are distributive up to coherent natural isomorphisms and is thus intrinsically2-categorical. The very different but essentially combinatorial ways around thisfound nonequivariantly in [8, 34] do not appear to generalize equivariantly, or atleast not easily. Our work involves conceptual categorical processing of 2-categoricalinput so that it feeds into an equivariant version of the 1-categorical Segal machine,whose multiplicative properties we have established in [15].An equivariant version of the Barratt-Priddy-Quillen theorem, which expressesthe suspension G -spectrum of a G -space as the equivariant algebraic K -theory ofa G -category, was proven in [13] using the equivariant operadic machine. However,this equivalence does not a priori preserve the multiplicative structure coming fromthe smash product of based G -spaces. The main result of [12] relies on having amultiplicative equivariant K -theory machine starting at the level of G -categories A much earlier operadic machine with target Lewis-May G -spectra [25] was developed byHauschild, May, and Waner. It was never published, but is outlined by Costenoble and Waner [6]. B. J. GUILLOU, J. P. MAY, M. MERLING, AND A. M. OSORNO that is compatible with the Barratt-Priddy-Quillen theorem, and we provide thatin this paper. An easier multiplicative version of the equivariant Barratt-Priddy-Quillen theorem is proven in [15, Theorem 6.7]), but that starts from categoricalinput that is quite different from the input needed in [12].We start with an equivariant K -theory machine K G producing orthogonal G -spectra from structured G -categories, which we take to be algebras over a suitableoperad O . In the nonequivariant case, the input would be permutative categories,which are algebras over the Barratt-Eccles operad. Conceptually, we would liketo extend K G to a monoidal functor from structured G -categories to orthogonal G -spectra. However, the ring G -categories that arise in nature are not the monoids fora monoidal structure on structured G -categories. Rather, as in [8, 24] and elsewhere,we have a multicategory structure on structured G -categories. A multicategorystructure on a category C allows one to make sense of the notion of monoid in C aswell as module over a monoid. We will thus extend K G to a multifunctor, meaningthat it is compatible with the multicategory structure.We give some intuition for finding the structure on an operad that ensures that itscategory of algebras is a multicategory. We think of the operad O as parametrizingaddition. Now suppose that we want to define a multiplication that distributes overaddition. Just as the product of integers mn is the n -fold addition of the integer m ,we can define a pairing O ( m ) × O ( n ) −→ O ( mn ) by repeating n times the variablein O ( m ) and then “adding” using the operad structure map. The diagram thatwe obtain when we compare this with the map O ( n ) × O ( m ) −→ O ( nm ) that weget by twisting in the source and using a reordering permutation in the target doesnot strictly commute in general. We define a pseudo-commutative operad to beone for which this comparison diagram commutes up to natural isomorphism (seeDefinition 3.10), and we show that this condition allows us to define a multicategorystructure on the category of O -algebras. A key example is the permutativity operad P G of [16, Definition 3.4]; its algebras are the permutative G -categories and itspseudoalgebras are the symmetric monoidal G -categories [16]. The operad P G is acategorical E ∞ G -operad as defined in [13, Definition 2.1], and when G = e it isjust the categorical E ∞ Barratt-Eccles operad.We write G U for the category of G -spaces and Cat ( G U ) for the 2-category ofcategories internal to G U , as described in Section 2.1. Fix a chaotic (Definition 2.5) E ∞ G -operad O in Cat ( G U ). We construct a multicategory Mult ( O ) whose un-derlying category is the category O - Alg ps of O -algebras and pseudomorphisms.Writing Sp G for the category of orthogonal G -spectra, we construct a functor K G : O - Alg ps −→ Sp G that group completes the additive structure, and most of the paper is devoted toestablishing the following result, which appears as Theorem 9.13. Theorem A.
Let O be a chaotic E ∞ G -operad in Cat ( G U ) . Then the functor K G : O - Alg ps −→ Sp G extends to a multifunctor. We have the following direct corollary of Theorem A.
Corollary. If A is a monoid in O - Alg ps , then K G ( A ) is a ring G -spectrum. If B is an A -module in O - Alg ps , then K G ( B ) is a K G ( A ) -module G -spectrum. ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 5 We warn the reader, however, that Theorem A does not assert that K G is sym-metric . In particular, we do not claim that our version of K G produces commu-tative ring G -spectra as output. Constructing a symmetric equivariant K -theorymultifunctor is an ongoing challenge. Our multifunctor K G is a composite of multi-functors all but one of which are symmetric, and we shall keep track of symmetryas we go along. However, associative and unital multiplicative properties are allthat are needed for the following result, which is the heart of what is needed in[12]. We prove the following theorem in Section 10. Here we use that G U embedsin Cat ( G U ), as recalled from [16, Remark 1.8] in Remark 9.20. Theorem B (Multiplicative equivariant Barratt-Priddy-Quillen) . Let O be a topo-logically discrete chaotic E ∞ G -operad in Cat ( G U ) and O + the associated monad.There is a lax monoidal natural transformation α : Σ ∞ G + −→ K G O + of functors G U −→ Sp G such that α X is a stable equivalence of orthogonal G -spectrafor all input G -CW complexes X . The main result of [12] gives a Quillen equivalence between the category of or-thogonal G -spectra and the category of “spectral Mackey functors,” i.e., spectrallyenriched functors G A −→ Sp , where G A is a spectral version of the Burnsidecategory. The proof of that result rests on having a multiplicative machine K G which satisfies Theorem B. Its construction was deferred to this paper. An al-ternative ∞ -categorical perspective on spectral Mackey functors as a model for G -spectra is given in [1, 4, 41]. Moreover, a version of the multiplicative equivar-iant Barratt-Priddy-Quillen theorem appears as [1, Theorem 10.6]. As the inputfor their machine differs from that of ours, a direct comparison of their result withours would be nontrivial but worthwhile. Remark 1.1.
An illuminating ∞ -category treatment of multiplicative infinite loopspace theory is given in [11]. We briefly compare that approach to the theory here.The input with that approach is symmetric monoidal ∞ -categories, which are ∞ -categorical generalizations of Segal’s special Γ-spaces. In this paper, as classically,the input is symmetric monoidal 1-categories, but we need to work with the 2-category of such, in order to keep track of the multiplicative structure. The focusof this paper is the passage from there to special Γ-categories, while the machine S G ◦ B from special Γ-categories to Ω- G -spectra is taken as a black box. We view themachine S G ◦ B as essentially formal. Like the ∞ -category machine, it is symmetricmonoidal, at least in the variant form given in [15]. Philosophically, from the ∞ -category point of view, we are showing that, even equivariantly, the passage fromsymmetric monoidal 1-categories to symmetric monoidal ∞ -categories preservesmultiplicative structure, albeit with a loss of symmetry.1.1. A road map.
The organization of this paper focuses on the multiplicativeelaboration of the following diagram, which displays K G as the composite of asequence of functors. B. J. GUILLOU, J. P. MAY, M. MERLING, AND A. M. OSORNO (1.2) O - Alg ps R (cid:15) (cid:15) K G / / Sp G D - Alg ps P (cid:15) (cid:15) F G - Top S G O O D G - Alg ps ζ ∗ G / / F G - PsAlg St / / F G - Alg B O O The notation
Alg ps denotes 2-categories of strict algebras and pseudomorphisms between them. Modulo just a bit of additional notational complexity, we can justas well replace the 2-categories (-)- Alg ps in the left hand column by more general2-categories (-)- PsAlg of pseudoalgebras and pseudomorphisms, with no significantchange in constructions or proofs. We shall often write R G for the composite PR in the left column.After a few preliminaries setting up our categorical framework of operads andmulticategories in Section 2, multicategories with underlying categories of the form O - Alg ps are defined in Section 3. Here O is a chaotic operad or, a bit more gen-erally, a pseudo-commutative operad. We develop a general categorical frameworkthat will specialize to an understanding of categories of operators over both finitesets and finite G -sets, together with their algebras and pseudoalgebras, in Section 4.We do this in a general framework that will later clarify some key distinctions. Mul-ticategories with underlying categories of the form D - Alg ps are defined in Section 5,where D is a category of operators over the category F of finite sets. Categories ofoperators were first introduced in [37], where they mediated between the operadicand Segalic infinite loop space machines. They play the same role here.Taking D to be the category of operators associated to a chaotic operad O , thefunctor R is constructed as a multifunctor in Section 5.5, but with a key proofdeferred to Section 12. All of this works in a general categorical context that apriori has nothing to do with either equivariance or topology. A crucial technicalpoint is that the “pseudo-commutative pairing” on an operad that we have alreadymentioned gives rise to an analogous “pseudo-commutative pairing” on its categoryof operators. The term “pseudo-commutative” was first coined by Hyland andPower [18] in a monadic avatar of our categories of operators; it can be viewed asshorthand for “pseudo-symmetric strict monoidal”.Working equivariantly, but in fact in a specialization of our general categoricalframework, multicategories with underlying categories of the form D G - Alg ps , where D G is a category of operators over the category F G of finite G -sets, are definedin Section 6. Taking D G to be the category of operators associated to a chaoticoperad O , the multifunctor P is also defined in that section.The categories of operators D and D G come with projections ξ : D −→ F and ξ G : D G −→ F G . Pulling back structure along these projections gives functors ξ ∗ and ξ ∗ G that send F -algebras to D -algebras and F G -algebras to D G -algebras, andsimilarly for pseudoalgebras. Taking full advantage of the equivariant context, we ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 7 construct a section ζ G : F G −→ D G to ξ G in Section 7. Pulling back along ζ G givesthe functor ζ ∗ G .However, since ζ G does not preserve structure as strictly as one might hope, ζ ∗ G takes strict algebras to pseudoalgebras. As we explain in Section 8, St is aspecialization of a general strictification functor due to Power and Lack [42, 20]that rectifies the loss of strictness and lands us in the multicategory associatedto the symmetric monoidal 2-category F G - Alg of strict F G -algebras in categoriesinternal to G -spaces and strict maps between them. Specializing general theorydeveloped in [16, 14], we explain in Section 8.2 how it extends to a multifunctor.From here, B is the standard classifying space functor and S G is the spacelevel multiplicative equivariant infinite loop space machine of [39]; F G - Top is thecategory of F - G -spaces and Sp G is the category of orthogonal G -spectra. We usethese functors to complete the proof of Theorem A in Section 9, and we combineour results here with results of [39] and [13] to prove Theorem B in Section 10.All of the multifunctors in (1.2) are symmetric except St and S G . We couldequally well have used the slightly more elaborate but equivalent choice for S G constructed in [15], which is symmetric. However, although ζ ∗ G is itself symmetric,loss of strict structure along it engenders the loss of symmetry of St, as we shallexplain in Section 8.3. Remark 1.3.
We alert the reader to an alternative route to Theorems A and B thatwas found at the same time as the one presented here. It will be presented in [28].It is illuminating, but it is more categorically intensive since it focuses on 2-monads,which we have avoided here despite this being a paper that is intrinsically all aboutthem. We will see in [28] that the k -ary morphisms in our operadic multicategoriesare the pseudoalgebras over a 2-monad M k and that the M k form a graded comonoidof 2-monads. Such structure also appears in other multicategorical contexts.The alternative route uses a 2-monadic reinterpretation of the vertical arrows in(1.2), but it replaces the horizontal composite St ◦ ζ ∗ G by a multifunctor whose under-lying map of 2-categories is the composite of Power-Lack strictification St : D G - Alg ps −→ D G - Alg and a derived variant of the left adjoint ξ G ∗ : D G - Alg −→ F G - Alg tothe forgetful functor ξ ∗ G : F G - Alg −→ D G - Alg . The section ζ ∗ G : D G - Alg −→ F G - PsAlg is a categorical shortcut that avoids use of ξ G ∗ , whose homotopical be-havior is problematic. The alternative route avoids any use of pseudoalgebras over F or F G , but we again lose symmetry, now due to the passage from ξ G ∗ to a homo-topically well-behaved derived variant. Conceivably, a more sophisticated derivedvariant might circumvent this.1.2. Acknowledgements.
This project has taken a long rocky road, and we havemany people to thank, too many to do justice to any of them. We are happy tothank Clark Barwick, Andrew Blumberg, Anna Marie Bohmann, David Gepner,Nick Gurski, Mike Hill, Akhil Mathew, Niko Naumann, Thomas Nikolaus, EmilyRiehl, David Roberts, Jonathan Rubin, Stefan Schwede, Michael Shulman, andDylan Wilson. We apologize to anyone we may have forgotten.
B. J. GUILLOU, J. P. MAY, M. MERLING, AND A. M. OSORNO Preliminaries on operads and multicategories
We begin here by introducing our categorical framework. We also recall thenotions of operads, their algebras, and pseudomorphisms between those. Finally,we recall the notion of a multicategory.
Notation 2.1.
Throughout the document, we will denote pseudomorphisms ofvarious types (for example, see Definition 2.11 or Definition 4.5) by arrows / / /o/o/o .2.1. V -categories. The categorical framework we begin with is the same as theone explained in more detail in [16, Section 1], hence we shall be brief.
Assumption 2.2.
We let V be a cartesian closed, bicomplete category.The examples of primary interest are V = U or V = G U , where U is the categoryof (compactly generated weak Hausdorff) spaces and G U is the category of G -spacesand G -maps for a finite group G . The reader focused on topology is free to read V as U , but nothing before Section 7 (or after Section 10) would change in any way.We defer further discussion of the equivariant context to Section 6.As discussed in more detail in [16, Section 1.1], we let Cat ( V ) denote the 2-category of categories, functors, and natural transformations internal to V . Wewill refer to these as V -categories , V -functors , and V -transformations . Thus any V -category C consists of objects Ob C and Mor C in V , and source, target, identity,and composition structure maps, which are all required to be morphisms in V . For C and D in Cat ( V ), a V -functor C −→ D is given by morphisms Ob C −→ Ob D and Mor C −→ Mor D in V that are suitably compatible with the internal categorystructure. A V -transformation α between V -functors F , F : C ⇒ D is given by amorphism α : Ob C −→ Mor D in V that makes the naturality diagrams commute.Since V is complete, so is Cat ( V ). Assumption 2.3.
We assume that
Cat ( V ) is moreover cocomplete.This assumption holds if either V is locally presentable or if V = U [49, (3.24)and (3.25)]. For similar reasons, it is also true for V = G U . Remark 2.4.
We note that
Cat ( V ) is cartesian closed since V is assumed to becartesian closed [19, Lemma 2.3.15]. Definition 2.5.
We say that a V -category C is chaotic , or indiscrete, if the sourceand target maps yield an isomorphism Mor C ( S,T ) −−−→ Ob C × Ob C . Chaotic V -categories and their properties are discussed in detail in [16, § Based V -categories. Let ∗ denote the terminal object of V . A basepoint ofan object V of V is a map ∗ −→ V in V . Write V ∗ for the category of based objectsof V and based maps. Let ∗ also denote the V -category whose object and morphismobjects are both given by the object ∗ of V . ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 9 Definition 2.6. A based V -category , or V ∗ -category , is a category internal to V ∗ .Equivalently, it is a category C internal to V equipped with a V -functor ∗ −→ C . Itsstructure maps source, target, identity, and composition must be in V ∗ . There arecorresponding notions of based functors, called V ∗ -functors , namely V -functors com-patible with basepoints, and based V -transformations, called V ∗ -transformations ,whose component morphisms Ob C −→ Mor D are based. As noted in [16, Re-mark 1.6], the resulting 2-category, here denoted Cat ( V ∗ ), can be identified with Cat ( V ) ∗ . Remark 2.7.
For a V -category or V ∗ -category C , an object of C will mean a functor ∗ −→ C or, equivalently, a morphism ∗ −→ Ob C in V . We warn the reader thatwe are using the term “object” in a technical sense. For example, when V is thecategory of G -spaces, an object is a G -fixed point of the G -space Ob C , hence C may have no objects.We can form the wedge and smash product of based V -categories A and B viathe pushout diagrams ∗ / / (cid:15) (cid:15) A (cid:15) (cid:15) ✤✤✤ B / / ❴❴❴ A ∨ B and A ∨ B / / (cid:15) (cid:15) A × B (cid:15) (cid:15) ✤✤✤ ∗ / / ❴❴❴❴ A ∧ B just as for spaces. Since the objects functor Ob : Cat ( V ) −→ V has both a left anda right adjoint and therefore preserves limits and colimits, it follows that(2.8) Ob ( A ∧ B ) ∼ = Ob ( A ) ∧ Ob ( B )for V ∗ -categories A and B .By the universal property of the smash product, a V ∗ -functor A ∧ B −→ C corresponds to a V -functor A × B −→ C whose restriction to ∗ × B and A × ∗ is the constant functor at the basepoint of C . This will allow us to define mapsfrom smash products by specifying basepoint conditions on V -functors defined onproducts.Similarly, a V ∗ -transformation A ∧ B F & & G ✤✤ ✤✤ (cid:11) (cid:19) ϕ C corresponds to a V -transformation of functors defined on A × B whose restrictionto A × ∗ and ∗ × B is the identity. Remark 2.9.
Our standing assumptions on V imply that Cat ( V ∗ ) is closed sym-metric monoidal with internal hom adjoint to ∧ (see [9, Lemma 4.20], [43, Con-struction 3.3.14]). We will use the symmetric monoidal structure to enrich categories over
Cat ( V ∗ )starting in Section 4. The associativity of ∧ is not formal from the universal property of the pushout and requires Cat ( V ) to be closed. In our applications, categories often have disjoint base objects, and we write A + for the coproduct (disjoint union in the relevant examples) of ∗ with an unbased V -category A . Then A + ∧ B + ∼ = ( A × B ) + (see [43, Lemma 3.3.16]).2.3. Operads in Cat ( V ) . We will work throughout with a reduced operad O in Cat ( V ), reduced meaning that O (0) is the trivial category ∗ . We will often assumethat O is chaotic, meaning that each V -category O ( n ) is chaotic. We will use thenotation γ : O ( k ) × O ( j ) × · · · × O ( j k ) −→ O ( j + · · · + j k )for the operad structure V -functors and : ∗ −→ O (1) for the unit object in O (1). Definition 2.10. An O -algebra is an object A in Cat ( V ) equipped with action V -functors θ ( n ) : O ( n ) × A n −→ A that are appropriately Σ n -equivariant, unital, and associative, as in [29]. Since O isassumed to be reduced, the functor θ (0) : ∗ −→ A specifies a basepoint 0 = 0 A ∈ A .We will be using non-strict maps between O -algebras, called O -pseudomorphisms.The full definitions of these and of O -transformations between them are given in[5] and, with some minor emendations, in [16, Definitions 2.23 and 2.24]. We shallnot repeat details, but we remind the reader of the key features. Definition 2.11.
Let A and B be O -algebras. An O -pseudomorphism A / / /o/o/o B is a V -functor F : A −→ B such that F (0 A ) = 0 B , together with invertible V -transformations ∂ n O ( n ) × A nθ ( n ) (cid:15) (cid:15) id × F n / / ✎✎✎✎ (cid:3) (cid:11) ∂ n O ( n ) × B nθ ( n ) (cid:15) (cid:15) A F / / B for n ≥ ∂ and the restriction of ∂ along × id : A ∼ = ∗ × A −→O (1) × A are identity V -transformations and such that the appropriate equality ofassociativity pasting diagrams relating the ∂ n to the structure maps of the operadholds (see [16, Definition 2.23]). It is a (strict) O -map if the ∂ n are identity V -transformations. Definition 2.12. An O -transformation between O -pseudomorphisms E and F isa V -transformation ω : E = ⇒ F such that the equality O ( n ) × A nθ ( n ) (cid:15) (cid:15) id × E n / / ✡✡✡✡ (cid:1) (cid:9) ∂ n O ( n ) × B nθ ( n ) (cid:15) (cid:15) A E ) ) F ✤✤ ✤✤ (cid:11) (cid:19) ω B = O ( n ) × A nθ ( n ) (cid:15) (cid:15) id × E n , , id × F n ✤✤ ✤✤ (cid:11) (cid:19) id × ω n ✎✎✎✎ (cid:3) (cid:11) ∂ n O ( n ) × B nθ ( n ) (cid:15) (cid:15) A F / / B holds for all n . We do not require the ω to be invertible. ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 11 Notation 2.13.
We will work throughout with the 2-category O - Alg ps of O -algebras, O -pseudomorphisms, and O -transformations.There is a more general definition of O -pseudoalgebras, as defined in [5, 16], butwe choose not to introduce it since it is not needed for the purposes of this paper.2.4. Review of multicategories.
We shall not repeat the complete definition ofa multicategory given in such sources as [8, 24, 52]. A multicategory M has a class Ob ( M ) of objects and for each sequence a = { a , . . . , a k } of objects, where k ≥ b , it has a set of k -ary morphisms M k ( a ; b ) = M k ( a , . . . , a k ; b ) . A quintessential example is that of k -linear maps in the category of vector spaces,which is why k -ary morphisms in arbitrary multicategories are sometimes called k -linear maps, even when there is no linear structure in sight.Throughout, we understand multicategories to be symmetric, so that the sym-metric group Σ k acts from the right on the collection of k -ary morphisms via maps σ : M k ( a , . . . , a k ; b ) −→ M k ( a σ (1) , . . . , a σ ( k ) ; b ) . For each object a there is an identity 1-ary morphism a −→ a and there are com-position functions(2.14) γ : M k ( b ; c ) × M j ( a ; b ) × · · · × M j k ( a k ; b k ) −→ M j ( { a , . . . , a k } ; c ) , where b is a k -tuple, a q for 1 ≤ q ≤ k is a j q -tuple, and, with j = j + · · · + j k , { a , . . . , a k } is the j -tuple { a , , . . . , a ,j , . . . , a k, , . . . , a k,j k } .The γ are subject to direct generalizations of the associativity, identity, andequivariance properties required of an operad in [29]. These properties are spelledout diagrammatically in [8, 2.1] and, with exceptional care, in [52, Chapter 11]. All of our multicategories are enriched in
Cat , but since that is only used pe-ripherally we will not go into detail. A multicategory with one object is then thesame thing as an operad in
Cat . Multicategories are often called colored operads,with objects thought of as colors. The objects and 1-ary morphisms of a multicate-gory M specify its underlying category, which is often also denoted M by abuse ofnotation. Remark 2.15.
There is a canonical multicategory Mult ( C ) associated to a sym-metric monoidal category ( C , ⊗ ). Its objects are those of C , and Mult k ( C )( a , . . . , a k ; b ) = C ( a ⊗ · · · ⊗ a k , b ) . It has the evident symmetric group actions and units. In schematic elementwisenotation, using the notations of (2.14), the composite of a k -ary morphism F : b −→ The colored operads in [52] are symmetric multicategories with a set of objects, called colors,but the generalization to a class of objects is evident. In fact, they are enriched in
Cat ( V ) when V is closed. There is a slight subtlety here. It has been said that there is a choice of such multicategoriesdepending on the chosen order of associating variables. With an unbiased operadic definition ofa symmetric monoidal category, the specification of
Mult ( C ) is unambiguous. c with ( E , . . . , E k ), where E r : a r −→ b r is a j r -ary morphism for 1 ≤ r ≤ k , is thecomposite(2.16) N ≤ r ≤ k N ≤ s ≤ j r a r,s ⊗ r E r / / N ≤ r ≤ k b r F / / c. This generalizes the example in vector spaces where k -linear maps correspond tomaps out of the tensor product.A morphism F : M −→ N of multicategories, called a multifunctor, is a function F : Ob ( M ) −→ Ob ( N ) together with functions F : M k ( a , . . . , a k ; b ) −→ N k ( F a , . . . , F a k ; F b )for all objects a i and b such that F (id a ) = id F ( a ) and F preserves composition. Ifthese functions are Σ k -equivariant, we say that F is a symmetric multifunctor . Alax monoidal (resp. lax symmetric monoidal) functor between symmetric monoidalcategories gives rise to a multifunctor (resp. symmetric multifunctor) between thecorresponsing multicategories.Given a multicategory M , one can define the notion of monoid in M (see [24, Ex-ample 2.1.11] or [52, § M is given by a multifunctor out of the appropriate param-eter multicategory into M . One can similarly define the notion of module over amonoid (see [8, Definition 2.5]). These notions agree with the usual ones whendealing with the multicategory associated to a symmetric monoidal category as inRemark 2.15. A multifunctor preserves associative and unital algebraic structures,and a symmetric multifunctor moreover preserves commutative ones.3. The multicategory of O -algebras The goal of this section is to establish a multiplicative structure on the category O - Alg ps of algebras over an operad. After some initial setup in Section 3.1, weintroduce the key concept of a pseudo-commutative operad in Section 3.2, follow-ing Corner and Gurski [5]. We then establish a multicategory Mult ( O ) for anypseudo-commutative operad O in Section 3.3, following Hyland and Power [18],and describe some variants in Section 3.4. Finally, we show that the free O -algebrafunctor extends to a multifunctor in Section 3.5.3.1. The intrinsic pairing of an operad.
Surprisingly, the following elementarystructure implicit in the definition of an operad is central to our work. It is presentin any reduced operad O in any cartesian monoidal category W . Definition 3.1.
The intrinsic pairing ⊛ : ( O , O ) −→ O of an operad O is given bythe composites O ( j ) × O ( k ) id × ∆ j / / O ( j ) × O ( k ) j γ / / O ( jk ) , where γ is the structure map of the operad and j ≥ k ≥ ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 13 Thinking of γ as specifying additive structure, the “product” ⊛ is taking seriouslythat jk = k + · · · + k . Thus the intrinsic pairing is an operadic manifestation of thegrade school lesson that multiplication is iterated addition. Remark 3.2.
The intrinsic pairing is not a pairing of operads in the sense originallydefined in [32, 1.4]. For many operads occurring naturally in topology, such as thelittle cubes or Steiner operads, the intrinsic pairing appears to be of no real interest.However, as we shall see in Section 3.3, it appears naturally when trying to constructa multicategory of algebras over an operad.
Proposition 3.3.
Let O be an operad in a cartesian monoidal category W . Then O = ` j ≥ O ( j ) is a monoid in W with product ⊛ and unit the unit object ∈ O (1) .It has a zero object ∗ ∈ O (0) .Proof. The unit properties of an operad are γ ( ; x ) = x and γ ( x ; j ) = x for x ∈ O ( j ). These say that is a unit for O . The associativity of the pairing is aneasy diagram chase from the following special case of the associativity diagram for γ in the definition of an operad. O ( j ) × O ( k ) j × O ( ℓ ) jk γ × id / / ∼ = (cid:15) (cid:15) O ( jk ) × O ( ℓ ) jk γ ' ' ❖❖❖❖❖❖❖❖❖❖❖ O ( jkℓ ) O ( j ) × ( O ( k ) × O ( ℓ ) k ) j id × γ j / / O ( j ) × O ( kℓ ) j γ ♦♦♦♦♦♦♦♦♦♦♦ Since O is reduced, ∗ ∈ O (0) is a zero element. (cid:3) Consider the category Σ of sets n = { , . . . , n } and isomorphisms. It is bipermu-tative under disjoint union and cartesian product. To be precise, the two monoidalstructures, ⊕ and ⊗ , are given by sum and product at the level of objects. To ap-ply ⊕ to permutations σ ∈ Σ j and τ ∈ Σ k and regard the result as a permutationof the j + k letters j + k = { , . . . , j + k } , we are implicitly applying the evidentisomorphism ζ j,k : j + k −→ j ∐ k, then taking the disjoint union of σ and τ , and then applying ζ − j,k . That is, σ ⊕ τ isdefined by the commutative diagram(3.4) j + k σ ⊕ τ / / ζ j,k (cid:15) (cid:15) j + kj ∐ k σ ∐ τ / / j ∐ k. ζ − j,k O O Similarly, define(3.5) λ = λ j,k : jk −→ j × k to be the order-preserving bijection, where j × k is ordered lexicographically. Then, µ ⊗ ν is defined by the commutative diagram(3.6) jk µ ⊗ ν / / λ j,k (cid:15) (cid:15) jkj × k µ × ν / / j × k. λ − j,k O O Recall that the associative operad
Assoc is given by
Assoc ( n ) = Σ n . Then ⊗ gives the intrinsic pairing of Definition 3.1 on Assoc . Moreover, if we think of thegroups Σ j as categories with a single object and thus think of Assoc as an operadin
Cat , then
Assoc = Σ and the monoidal structure of Proposition 3.3 is given by ⊗ . In particular, e j ⊗ e k = e jk . Since ⊗ is a group homomorphism, it is equivariantin the sense that µσ ⊗ ντ = ( µ ⊗ ν )( σ ⊗ τ ). Clearly e ⊗ ν = ν and µ ⊗ e = µ . Remark 3.7.
The equivariance formulas for an operad O imply that the pairingon Σ and the pairing on O are compatible in the sense that for all σ ∈ Σ j and τ ∈ Σ k the following diagram commutes. O ( j ) × O ( k ) ⊛ / / σ × τ (cid:15) (cid:15) O ( jk ) σ ⊗ τ (cid:15) (cid:15) O ( j ) × O ( k ) ⊛ / / O ( jk ) Definition 3.8.
Let τ j,k ∈ Σ jk be the permutation specified by the composite jk λ j,k / / j × k t / / k × j λ − k,j / / kj = jk It reorders the set j × k from lexicographic ordering to reverse lexicographic ordering.Clearly τ − j,k = τ k,j and τ ,n = e n = τ n, .The τ i,j are the symmetry isomorphisms for ⊗ in Σ. More precisely, for µ ∈ Σ j and ν ∈ Σ k , we have the commutative diagram jk τ j,k (cid:15) (cid:15) λ j,k / / j × k µ × ν / / t (cid:15) (cid:15) j × k t (cid:15) (cid:15) λ − jk / / jk τ j,k (cid:15) (cid:15) kj λ k,j / / k × j ν × µ / / k × j λ − k,j / / kj. That is,(3.9) τ j,k ( µ ⊗ ν ) = ( ν ⊗ µ ) τ j,k or equivalently ( µ ⊗ ν ) τ k,j = τ k,j ( ν ⊗ µ ) . Pseudo-commutative operads.
Recall the permutativity operad P from[13, Definition 4.1]. It is the chaotic categorification of Assoc , and as an operadin
Cat , its algebras are in one-to-one correspondence with permutative categories.
ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 15 The intrinsic pairing ⊛ on P is inherited from that of Assoc . Note that equation(3.9) implies that the diagram P ( j ) × P ( k ) t (cid:15) (cid:15) ⊛ / / P ( jk ) τ k,j (cid:15) (cid:15) P ( k ) × P ( j ) ⊛ / / P ( kj )does not in general commute. Rather, since P ( kj ) is chaotic, there exists a naturalisomorphism α j,k : τ k,j ◦ ⊛ = ⇒ ⊛ ◦ t . This is an example of a pseudo-commutativeoperad, as defined by Corner and Gurski [5, § Definition 3.10.
Let O be an operad in Cat ( V ). A pseudo-commutative structure on an operad O is a collection of invertible V -transformations, one for each ( j, k ),of the form(3.11) O ( j ) × O ( k ) ⊛ / / t (cid:15) (cid:15) ✌✌✌✌ (cid:2) (cid:10) α j,k O ( jk ) τ k,j (cid:15) (cid:15) O ( k ) × O ( j ) ⊛ / / O ( kj ) . The α j,k must satisfy coherence axioms for identity, symmetry, equivariance, andoperadic compatibility that are specified and discussed in Section 11.1.If O is chaotic, such transformations α always exist and all conditions are auto-matically satisfied [16, § Lemma 3.12. ( [5, Corollary 4.9] ) A chaotic operad has a unique pseudo-com-mutative structure. We will often write “pseudo-commutative operad” when we really mean “operadequipped with a pseudo-commutative structure”, but there is no ambiguity when O is chaotic, and we later prefer to specialize to chaotic operads.3.3. The multicategory of O -algebras. Hyland and Power [18] show that thereis a multicategory of algebras over a pseudo-commutative monad, and Corner andGurski show in [5] that the monad corresponding to a pseudo-commutative operadis pseudo-commutative in the sense of [18]. We follow these sources to describe themulticategory of algebras over a pseudo-commutative operad.For the pairings of O -algebras we want to consider maps F : A × B −→ C thatpreserve the algebra structure on each variable up to canonical isomorphism. For The original definition of [5] requires some minor corrections that are given there. example, if + denotes a binary operation in O , we need to make sense of a distribu-tivity law of the general form F ( a, b + b ) ∼ = F ( a, b ) + F ( a, b ) . Diagonal maps enter since a appears once on the left and twice on the right.The definition contains a number of schematic coherence diagrams to the effectthat whenever two natural transformations have a chance to be equal they areequal. We shall explain the diagrams after giving the definition. The followingmaps s i play a key role. Notation 3.13.
Let A i , 1 ≤ i ≤ k , be V -categories and let n ≥
0. Define s i to bethe composite V -functor displayed in the diagram A × · · · × A i − × O ( n ) × A ni × A i +1 × · · · × A k s i / / t ∼ = (cid:15) (cid:15) O ( n ) × ( A × · · · × A k ) n O ( n ) × A × · · · × A i − × A ni × A i +1 × · · · × A k . id × ∆ / / O ( n ) × A n × · · · × A nk ∼ = O O Here t is the evident transposition, ∆ is obtained by applying the diagonal maps A j −→ A nj for j = i , and the right hand isomorphism is obtained by transposingfrom a product of n th powers to an n th power of a product. Definition 3.14.
Let O be a (reduced) pseudo-commutative operad in Cat ( V ).We define the (symmetric) multicategory Mult ( O ) of O -algebras and pseudomor-phisms. Its underlying 2-category is O - Alg ps , so its objects, morphisms, and2-cells are the O -algebras, the O -pseudomorphisms (Definition 2.11), and the O -transformations (Definition 2.12). Recall that since O is reduced, all O -algebrasare assigned basepoints. Its 0-ary morphisms are (unbased) maps ∗ −→ B , thatis, they correspond to a choice of object in B . For k >
1, its k -ary morphisms( A , . . . , A k ) −→ B are the tuples ( F, δ i ), where(a) F : A ∧ · · · ∧ A k −→ B is a V ∗ -functor, which we may equally well express asa V -functor F : A × · · · × A k −→ B such that F ( a , . . . , a k ) is equal to 0 B ifany object a i is 0 A i and F ( f , . . . , f k ) is id B if any f i is id A i , and(b) the δ i , 1 ≤ i ≤ k , are sequences of invertible V -transformations δ i ( n ) as indi-cated in the following diagram.(3.15) O ( n ) × ( A × · · · × A k ) n id × F n / / ✏✏✏✏ (cid:4) (cid:12) δ i ( n ) O ( n ) × B nθ ( n ) (cid:15) (cid:15) A × · · · × O ( n ) × A ni × · · · × A ks i O O id × θ ( n ) × id (cid:15) (cid:15) A × · · · × A k F / / B The elementary maps s i correspond to the “strengths” t i in Hyland and Power [18, p. 156];in their categorical treatment, the existence of t i with suitable properties is an axiom on a given2-monad, although they do make the strengths explicit in the case of permutative categories. ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 17 The distributivity isomorphisms δ i ( n ) must satisfy coherence axioms that are spec-ified and discussed in Section 11.2.For σ ∈ Σ k , the right action of Σ k on the k -ary morphisms of Mult ( O ) sends( F, δ i ) : ( A , . . . , A k ) −→ B to the composite(3.16) A σ (1) × · · · × A σ ( k ) σ / / A × · · · × A k F / / B , where σ denotes the reordering of terms given by σ ( a σ (1) , . . . , a σ ( k ) ) = ( a , . . . , a k ) . Permuting the indices, the δ i for F σ are inherited from the δ i for F . Precisely, δ σ − ( i ) ( n ) for F σ is induced from δ i ( n ) for F by pasting the defining diagram (3.15)to the right of the following diagram, which is easily checked to be commutative. O ( n ) × ( A σ (1) × · · · × A σ ( k ) ) n id × σ n / / O ( n ) × ( A × · · · × A k ) n A σ (1) × · · · × O ( n ) × A ni × · · · × A σ ( k ) s σ − i ) O O id × θ ( n ) × id (cid:15) (cid:15) σ / / A × · · · × O ( n ) × A ni × · · · × A ks i O O id × θ ( n ) × id (cid:15) (cid:15) A σ (1) × · · · × A σ ( k ) σ / / A × · · · × A k Since i = σσ − ( i ), the term O ( n ) × A ni appears in the σ − ( i )th factor of the middleleft term. Note that ( F σ ) τ = F ( στ ), both mapping A στ (1) × · · · × A στ ( k ) to B .The identity functor of A gives the unit element id A ∈ Mult ( O )( A ; A ). Withthe notation for sequences from Section 2.4, the composition multiproduct Mult ( O )( B ; C ) × Q kq =1 Mult ( O )( A q ; B q ) γ / / Mult ( O )( {A , . . . A k } ; C )is given by γ ( F ; E , . . . , E k ) = F ◦ ( E ∧ · · · ∧ E k ) . We identify { ( q, r ) } , 1 ≤ q ≤ k and 1 ≤ r ≤ j q , with { ≤ i ≤ j + · · · j k } by letting( q, r ) correspond to i = j + · · · + j q − + r . Then δ q,r ( n ) for the multicomposition isgiven by pasting the diagrams for δ Fq and δ E q r . We show this explicitly in the caseof ( q, r ) = (1 , A q forthe product A q, × · · · × A q,j q .(3.17) O ( n ) × (cid:0) A × · · · × A k (cid:1) n id × ( E ×···× E k ) n / / O ( n ) × ( B × · · · × B k ) n id × F n / / O ( n ) × C nθ ( n ) (cid:15) (cid:15) O ( n ) × A n × A × · · · × A ks O O id × E n × E ×···× E k / / ✒✒✒✒ (cid:5) (cid:13) δ E ( n ) × id O ( n ) × B n × B × · · · × B kθ ( n ) × id (cid:15) (cid:15) s O O ☞☞☞☞ (cid:2) (cid:10) δ F ( n ) O ( n ) × A n , × A , × · · · × A k,j k s × id O O θ ( n ) × id (cid:15) (cid:15) A × · · · × A k E ×···× E k / / B × · · · × B k F / / C One can check that the δ s satisfy the coherence axioms, and that this compositionis associative, unital and respects equivariance. Further verifications are needed toshow that this all really does specify a multicategory. For example, the symmetryaxiom (ii) in Section 11.1 is used in the verification that F σ satisfies the axiomswhen F does. However, we omit further details. We have translated the axiomsof Hyland and Power to our operadic setting. Their [18, Proposition 18] appliesto show that Mult ( O ) is a multicategory enriched in the category Cat of smallcategories. We learned the central role played by pseudo-commutativity from them.3.4.
Variants of Mult ( O ) and comparisons. We remark that our definition of
Mult ( O ) applies almost verbatim to define a multicategory of O -pseudoalgebrasas defined in [5, 16]. Pseudoalgebras over an operad are defined by relaxing thecoherence diagrams for the operadic multiplication with the structure map of thealgebra to only commute up to coherent natural isomorphisms. The only axiomsin the definition of O -pseudomorphisms (listed in Section 11.2) that would differslightly for O -pseudoalgebras as opposed to O -algebras are (iv) and (v). In theseconditions it wouldn’t make sense to ask for equality of 2-cells as written, sincethe maps on the boundary of the 2-cells would not be equal, and instead onewould need to paste them with the coherence isomorphisms from the definition of O -pseudoalgebras.When V = U and O is the permutativity operad P , the multicategory Mult ( P )is not quite the same as the multicategory of symmetric strict monoidal categoriesdefined by Hyland and Power [18] and the multicategory of permutative categoriesdefined by Elmendorf and Mandell [8]. The difference is that we have taken ourdistributivity 2-cells δ i to be invertible. Neither [18] nor [8] do so, and we havedrawn our arrows in the direction used in [18], which is opposite to the choice in[8]. This difference in the choice of direction of the 2-cells δ i would matter if we re-laxed the isomorphism requirement. For example, the strengths s i of Notation 3.13would no longer be relevant with the opposite choice, so the definition in [8] wouldno longer be a specialization of [18] and would not be compatible with the con-ventions of Corner and Gurski [5] or with LaPlaza’s classical coherence theory forsymmetric bimonoidal categories [21, 22]. It would therefore lead to some erroneousconclusions, as explained in [34, Scholium 12.3].The work of [8] used the classical biased definition of permutative categoriesrather than its unbiased operadic equivalent, and that simplifies details when com-paring operadic algebraic structures to their classical biased equivalents. With ourunbiased operadic reformulation, the equivariant generalization is immediate. Forexample, we can take O to be the categorical equivariant Barratt-Eccles operad P G of [13] to obtain the multicategory Mult ( P G ) of genuine permutative G -categories;genuine permutative and symmetric monoidal G -categories are defined to be P G -algebras and P G -pseudoalgebras, respectively, in [13, 16]. Our work also appliesto the normed symmetric monoidal categories of Rubin [45], which are defined asalgebras over an operad, but which also admit a biased definition.3.5. The free O -algebra multifunctor O + . As explained in [35, Section 4], a(reduced) operad O in a category W has two associated monads, O defined on theground category W ∗ and O + defined on the ground category W . Their categories ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 19 of algebras are isomorphic. The first takes the basepoint as given and requiresthe basepoint built in by the operad action to agree with the given one, and it isdefined using basepoint identifications. The second just builds in the basepoint bythe action. The first is the one central to topology and is in principle more general.For an unbased object X ∈ W ,(3.18) O + X = O ( X + ) = a j ≥ O ( j ) × Σ j X j . This is a based object with basepoint given by the inclusion of ∗ = O (0) × X .Starting on the category level with W = Cat ( V ), we prefer to avoid basepointidentifications and we therefore focus on O + . When V is U or G U , applying theclassifying space functor gives an operad B O , and we denote the associated monadby O top + . If O is Σ-free, in the sense that Σ j acts freely on O ( j ) for each j , we thenhave the basic commutation relation(3.19) B ( O + C ) ∼ = O top + ( B C )for C ∈ Cat ( V ), and that is essential to our applications.Thus fix a chaotic operad O in Cat ( V ) in this section. Definition 3.20.
For V -categories C and D , define a V ∗ -functor ω : O + C ∧ O + D −→ O + ( C × D )by passage to orbits from the maps(3.21) O ( j ) × C j × O ( k ) × D k t / / O ( j ) × O ( k ) × C j × D k ⊛ × ℓ / / O ( jk ) × ( C × D ) jk . The map ℓ here is defined using the lexicographic ordering λ of (3.5); explicitly, itis given by( c , . . . , c j ) , ( d , . . . , d k ) ( c , d ) , . . . ( c , d k ) , . . . . . . , ( c j , d ) , . . . , ( c j , d k ) . Since O is reduced, the maps (3.21) factor through the smash product and, usingthe equivariance axiom for operad composition, they also pass to orbits with respectto symmetric group actions; therefore they induce a well-defined map ω . Proposition 3.22.
A pseudo-commutativity structure on O induces an invertible V ∗ -transformation (3.23) O + C ∧ O + D ω / / t (cid:15) (cid:15) ✏✏✏✏ (cid:4) (cid:12) α O + ( C × D ) O + ( t ) (cid:15) (cid:15) O + D ∧ O + C ω / / O + ( D × C ) . Proof.
By axiom (iii) from Definition 3.10, the transformations α j,k descend toorbits. It is straightforward to check that they define the claimed invertible V ∗ -transformation. (cid:3) We now extend ω to a binary morphism in Mult ( O ). We need to define thetransformations δ i ( n ) for i = 1 ,
2. Careful inspection shows that we can take δ ( n ) to be the identity transformation. That is, we claim that the following diagramcommutes. O ( n ) × ( O + C × O + D ) n id × ω n / / O ( n ) × O + ( C × D ) nθ ( n ) (cid:15) (cid:15) O ( n ) × ( O + C ) n × O + D s O O θ ( n ) × id (cid:15) (cid:15) O + C × O + D ω / / O + ( C × D ) In particular, it is important to notice that the variables in C and D are arrangedlexicographically under either composite. The variables in the operad agree underthe composite by iterated application of the associativity diagram for the structuremaps γ , as in Proposition 3.3.We define δ ( n ) as the following pasting diagram(3.24) O ( n ) × ( O + C × O + D ) n id × ω n / / id × t n * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ ✔✔✔✔ (cid:6) (cid:14) id × α n O ( n ) × O + ( C × D ) nθ ( n ) (cid:15) (cid:15) id × O + ( t ) n u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ O ( n ) × ( O + D × O + C ) n id × ω n / / O ( n ) × O + ( D × C ) nθ ( n ) (cid:15) (cid:15) O + C × O ( n ) × ( O + D ) ns O O id × θ ( n ) (cid:15) (cid:15) t / / O ( n ) × ( O + D ) n × O + C s O O θ ( n ) × id (cid:15) (cid:15) O + D × O + C ω / / t t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ ✔✔✔✔ (cid:6) (cid:14) α O + ( D × C ) O + ( t ) ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ O + C × O + D ω / / O + ( C × D ) , where the inner pentagon is δ = id. Proposition 3.25.
The axioms on α imply that ( ω, δ = id , δ ) satisfies the con-ditions for a 2-ary morphism in Mult ( O ) given in Section 11.2.Proof. The most difficult axiom to verify is (v). The operadic compatibility condi-tion on α (Definition 3.10 (iv)) is central to this verification. We leave the detailsto the reader. (cid:3) For ease of notation we will denote this morphism by ω . The following resultfollows easily from the definition of ω . Lemma 3.26.
The pairing ω is natural, in the sense that for all V -functors F : A −→ C , H : B −→ D , the diagram ( O + A , O + B ) ω / / ( O + F, O + H ) (cid:15) (cid:15) O + ( A × B ) O + ( F × H ) (cid:15) (cid:15) ( O + C , O + D ) ω / / O + ( C × D ) in Mult ( O ) commutes. ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 21 Lemma 3.27.
Given V -categories C , D , and E , the following diagram in Mult ( O ) commutes. ( O + C , O + D , O + E ) ( ω, id) / / (id ,ω ) (cid:15) (cid:15) ( O + ( C × D ) , O + E ) ω (cid:15) (cid:15) ( O + C , O + ( D × E )) ω / / O + ( C × D × E ) Proof.
Since the intrinsic pairing ⊛ is associative, the diagram O + C ∧ O + D ∧ O + E ω ∧ id / / id ∧ ω (cid:15) (cid:15) O + ( C × D ) ∧ O + E ω (cid:15) (cid:15) O + C ∧ O + ( D × E ) ω / / O + ( C × D × E )commutes; the compatibility of the δ i follows from the conditions on α . (cid:3) Thus, given V -categories C , . . . , C k , we have a corresponding k -ary morphism ω k : ( O + C , . . . , O + C k ) −→ O + ( C × · · · × C k )in Mult ( O ), defined by using ω iteratively. Since this is a composition in Mult ( O ),the δ i for ω k are obtained from those for ω using the pasting (3.17). We take ω = idand take ω to be the choice of object ( , ∗ ) ∈ O (1) × ∗ ⊂ O + ( ∗ ). Theorem 3.28.
The functor O + from V -categories to O -algebras extends to amultifunctor O + : Mult ( Cat ( V )) −→ Mult ( O ) . We do not claim that the multifunctor we construct is symmetric, and we shallshow that it is not in Remark 3.29.
Proof. In Mult ( Cat ( V )), a k -ary morphism is just a V -functor F : C ×· · ·× C k −→ D . Its image under the multifunctor is the composite( O + C , . . . , O + C k ) ω k / / O + ( C × · · · × C k ) O + F / / O + D . It is clear that this assignment sends the identity of C to the identity of O + ( C ).The fact that the assignment preserves composition follows from the functorialityof O + and the naturality of ω (Lemma 3.26). (cid:3) Remark 3.29.
The multifunctor O + is not symmetric. To see this, consider id C × D as a bilinear map in Mult ( Cat ( V )). On one hand, if we first hit it with thesymmetry t and then O + , we end up with a bilinear map whose V ∗ -functor is givenby O + D ∧ O + C ω / / O + ( D × C ) O + ( t ) / / O + ( C × D ) . If, on the other hand, we first do O + and next t , we get O + D ∧ O + C t / / O + ( C ) × O + ( D ) ω / / O + ( C × D ) . These maps do not agree since they differ by the two-cell α . V ∗ - categories and their algebras and pseudoalgebras This section establishes terminology and notation that will be used frequentlyin the coming sections. Much of what we do in Section 4.1 is to describe explicitlywhat it means to do enriched category theory over the 2-category
Cat ( V ). InSection 4.2, we introduce algebras and pseudoalgebras in this context, and we setourselves up to discuss multiplicative structures on our categories of algebras byintroducing monoidal structures on our enriched categories in Section 4.3.4.1. V ∗ - -categories. All of our categories of operators, which we introduce inSections 5.1 and 6.2, are examples of V ∗ -2-categories, and we explain what thoseare here. Returning to Section 2, we note that Cat ( V ) and Cat ( V ∗ ) are symmetricmonoidal 2-categories, with cartesian and smash product, respectively. As such,we can enrich and weakly enrich over them, in the sense of [10]. Classically, a2-category is a category enriched in the category Cat of (small) categories.
Definition 4.1.
We refer to categories, functors, and natural transformations en-riched in
Cat ( V ) as V - -categories , V - -functors , and V - -natural transformations ,respectively. Similarly, we call categories, functors, and natural transformations en-riched in Cat ( V ∗ ) V ∗ - -categories , V ∗ - -functors , and V ∗ - -natural transformations ,respectively.We briefly unpack these definitions. A V -2-category C consists of a collection ofobjects (0-cells) and a morphism V -category C ( c, d ) for each pair of objects ( c, d )(giving objects in V of 1-cells and 2-cells). For a V ∗ -2-category we moreover havethat each C ( c, d ) has a basepoint, and the composition factors through the smashproduct. If C and D are V -2-categories, a V -2-functor F : C −→ D is given bya function F on objects and V -functors C ( c, d ) −→ D ( F ( c ) , F ( d )) satisfying theevident unit and associativity conditions (strictly). For a V ∗ -2-functor we furtherrequire that the latter are V ∗ -functors.Finally, if E and F are V -2-functors C −→ D , where C and D are V -2-categories,a V -2-natural transformation ζ : E = ⇒ F consists of 1-cells ζ c : E ( c ) −→ F ( c ),meaning objects of D ( E ( c ) , F ( c )), such that the naturality diagrams C ( c, d ) E / / F (cid:15) (cid:15) D ( E ( c ) , E ( d )) ( ζ d ) ∗ (cid:15) (cid:15) D ( F ( c ) , F ( d )) ( ζ c ) ∗ / / D ( E ( c ) , F ( d ))commute for all pairs ( c, d ) of objects of C . The same is true for a V ∗ -2-naturaltransformations, except that the diagram above lives in Cat ( V ∗ ).As in [39, Section 1.3], there is a close relationship between V -2-categories witha zero object and V ∗ -2-categories, which we now explain. For a V -2-category C , wesay that in C is a zero object if C ( c, d ) ∼ = ∗ if either c = or d = . If C has a As in Remark 2.7, a 1-cell here means a V -functor ∗ −→ D ( E ( c ) , F ( c )). ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 23 zero object, each hom V -category C ( c, d ) is based with basepoint(4.2) 0 c,d : ∗ ∼ = C ( , d ) × C ( c, ) ◦ / / C ( c, d ) . Proposition 4.3.
For C a V -2-category with zero object , the basepoints (4.2) give C an enrichment in Cat ( V ∗ ) .Proof. We only need to check that composition in C factors through the smashproduct. The associativity diagram C ( d, e ) × C ( , d ) × C ( c, ) id ×◦ / / ◦× id (cid:15) (cid:15) C ( d, e ) × C ( c, d ) ◦ (cid:15) (cid:15) ∗ ∼ = C ( , e ) × C ( c, ) ◦ / / C ( c, e )shows that composition sends C ( d, e ) × c,d to 0 c,e and the symmetric argumentshows that composition also sends 0 d,e × C ( c, d ) to 0 c,e . (cid:3) The following result, whose proof we leave to the reader, characterizes V ∗ -2-functors for V -2-categories with a zero object. Proposition 4.4. If C and D have zero objects, then V ∗ -2-functors F : C −→ D correspond bijectively to V -2-functors F : C −→ D that are reduced, in the sensethat F ( C ) ∼ = D . Moreover, V ∗ -2-natural transformations correspond bijectively to V -2-natural transformations between reduced V -2-functors (that is, there is no extracondition). Following [10, Sections 3.5 and 3.7], we now introduce the weaker notions of V -pseudofunctor and V -pseudonatural transformation, together with their basedvariants. Definition 4.5. A V -pseudofunctor F : C / / /o/o/o D between V -2-categories consistsof a function F on objects and V -functors F : C ( b, c ) −→ D ( F ( b ) , F ( c ))such that the following diagram commutes(4.6) ∗ id c / / id F ( c ) % % ❑❑❑❑❑❑❑❑❑❑❑ C ( c, c ) F (cid:15) (cid:15) D ( F ( c ) , F ( c )) , together with invertible coherence V -transformations(4.7) C ( b, c ) × C ( a, b ) ◦ (cid:15) (cid:15) ✕✕✕✕ (cid:6) (cid:14) ϕF × F / / D ( F ( b ) , F ( c )) × D ( F ( a ) , F ( b )) ◦ (cid:15) (cid:15) C ( a, c ) F / / D ( F ( a ) , F ( c )) In the language of [16], we are restricting to normal V -pseudofunctors. that are unital ( ϕ id , − and ϕ − , id are the identity) and associative in the sense thatthe relevant equalities of pasting diagrams relating to triple composition hold (seefor example [23, Section 1.1] or [10, Section 3.5]).The based variant is defined similarly. Definition 4.8. A V ∗ -pseudofunctor F : C / / /o/o/o D between V ∗ -2-categories is a V -pseudofunctor such that the functors on morphisms are V ∗ -functors, with the unitdiagram (4.6) replaced with the diagram of V ∗ -functors with source ∗ ∐ ∗ , andthe transformations ϕ in (4.7) descend to V ∗ -transformations of V ∗ -functors withsource C ( b, c ) ∧ C ( a, b ).That is, in the diagram (4.7) for a V ∗ -pseudofunctor, both instances of × in thetop row are replaced by ∧ . Definition 4.9.
Let E and F be V -pseudofunctors C / / /o/o/o D , where C and D are V -2-categories. Then a V -pseudotransformation ζ : E = ⇒ F consists of 1-cells ζ c : E ( c ) −→ F ( c ) for objects c ∈ C and invertible V -transformations C ( b, c ) F / / E (cid:15) (cid:15) ✑✑✑✑ (cid:4) (cid:12) ζ b,c D ( F ( b ) , F ( c )) ( ζ b ) ∗ (cid:15) (cid:15) D ( E ( b ) , E ( c )) ( ζ c ) ∗ / / D ( E ( b ) , F ( c ))for objects b, c ∈ C such that the component of ζ c,c at id c is the identity 2-cellfor all c ∈ C and the relevant coherence diagram expressing compatibility withcomposition commutes (see for example [23, Section 1.2] or [10, Section 3.7]).Again, the definition is essentially the same in the based context. Definition 4.10.
Let E and F be V ∗ -pseudofunctors C / / /o/o/o D , where C and D are V ∗ -2-categories. Then a V ∗ -pseudotransformation ζ : E = ⇒ F is a V -pseudotransformationof the underlying V -pseudofunctors such that each ζ b,c is a V ∗ -transformation, mean-ing that its component at 0 b,c is the identity.As in Proposition 4.4, we have the following characterization of V ∗ -pseudofunctors. Proposition 4.11. If C and D have zero objects, then V ∗ -pseudofunctors C / / /o/o/o D correspond bijectively to V -pseudofunctors C / / /o/o/o D that are reduced, in the sensethat F ( C ) ∼ = D , and are such that ϕ restricts to the identity on the subcategories b,c × C ( a, b ) and C ( b, c ) × a,b .Moreover, V ∗ -pseudotransformations ζ : E = ⇒ F correspond bijectively to V -pseudotransformations between the underlying V -pseudofunctors (that is, there isno extra condition). As in any enriched setting, we have the following construction. This is equivalent to requiring that the unit diagram (4.6) commutes as a diagram of under-lying V -functors. This is meant as an abbreviation of V -pseudonatural transformation. ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 25 Definition 4.12. If C and D are V -2-categories, their product V -2-category C × D has objects Ob ( C ) × Ob ( D ) and morphism V -categories( C × D )(( c, d ) , ( c ′ , d ′ )) = C ( c, c ′ ) × D ( d, d ′ ) , with the evident composition. If C and D have zero objects, so does C × D , withzero object ( , ).Similarly, if C and D are V ∗ -2-categories, we have a V ∗ -2-category C ∧ D withobjects Ob ( C ) × Ob ( D ) and with( C ∧ D )(( c, d ) , ( c ′ , d ′ )) = C ( c, c ′ ) ∧ D ( d, d ′ )as the V ∗ -category of morphisms.The following remark allows us to regard 2-categories as V -2-categories and es-tablishes a convenient context. Remark 4.13.
The underlying set functor U : V −→
Set specified by U X = V ( ∗ , X ) has left adjoint V : Set −→ V specified by V S = ` s ∈ S ∗ , the coproduct ofcopies of the terminal object ∗ indexed on the elements of the set S . Thus(4.14) Set ( S, U X ) ∼ = V ( V S, X ) . When V is strongly complete [16, Definition 6.2], a mild condition that holds in allrelevant examples, V preserves finite limits. We can apply this with V replaced by Cat ( V ), so that(4.15) Cat ( B , U X ) ∼ = Cat ( V )( V B , X ) . for a category B and a V -category X . We regard categories B as discrete 2-categories, meaning that they have only identity 2-cells. Applying V to the homcategories of 2-categories allows us to change their enrichment from Cat to Cat ( V ).Thus we may regard categories B as V -2-categories, and we agree to do so withoutchange of notation. We then call B a discrete V - -category . In our examples, B has a zero object and therefore gives rise to a V ∗ -2-category.4.2. Algebras and pseudoalgebras over V ∗ - -categories. Classically, algebrasover a category D enriched in based spaces can be defined as enriched functors X : D → U ∗ . The “action” of the category D can be seen by translating thisenriched functor into adjoint form as the data of a family of compatible continuousmaps D ( a, b ) ∧ X ( a ) → X ( b ) indexed by objects a, b of D . Similarly, algebrasover an operad O in U can be defined either via action maps O ( j ) × X j → X orequivalently via a map of operads from O to the endomorphism operad End( X ). Remark 4.16.
This second interpretation can always be given when the ambientcategory is closed, as we have assumed. Indeed, if V is closed, it has an internalhom object V ( V, W ) for each pair of objects and an adjunction V ( V × W, Z ) ∼ = V ( V, V ( W, Z )) . Each object V then has an endomorphism operad End( V ) withEnd( V )( j ) = V ( V j , V ) . An action of an operad O in V on an object V ∈ V is then the same as a map ofoperads O −→
End( V ). This gives an adjoint specification of O -algebras. As Cat ( V ) is cartesian closed (see Remark 2.4), we will write Cat ( V ) for theresulting V -2-category. Similarly, Cat ( V ∗ ) is closed by Remark 2.9, and we willwrite Cat ( V ∗ ) for the resulting V ∗ -2-category.If C is a V -2-category, it is natural to define a C -algebra X to be a V -2-functor X : C −→ Cat ( V ). If C has a zero object, we say that X is reduced if X ( ) = ∗ .We have the following enhancement of Propositions 4.4 and 4.11. Proposition 4.17.
Let C be a V -2-category with zero object. Then(1) V ∗ - -functors C −→ Cat ( V ∗ ) correspond bijectively to reduced V - -functors C −→ Cat ( V ) .(2) V ∗ -pseudofunctors C / / /o/o/o Cat ( V ∗ ) correspond bijectively to reduced V -pseudofunctors C / / /o/o/o Cat ( V ) such that ϕ restricts to the identity onthe subcategories b,c × C ( a, b ) and C ( b, c ) × a,b .Proof. To prove (1), let X : C −→ Cat ( V ) be a reduced V -2-functor. Then for c ∈ C ,the adjoint of the map C ( , c ) −→ Cat ( V )( X ( ) , X ( c ))endows X ( c ) with a basepoint: ∗ ∼ = C ( , c ) × X ( ) −→ X ( c ) . Functoriality can then be used to check that this indeed gives rise to a V ∗ -2-functor C −→ Cat ( V ∗ ). For the converse we apply Proposition 4.4 with D = Cat ( V ∗ ), andthen forget the basepoints to get a map with target Cat ( V ).The argument for (2) is similar, with the caveat that the condition on ϕ isnecessary to ensure that we do get a map with target Cat ( V ∗ ). (cid:3) Definition 4.18.
Let C be a V ∗ -2-category. A (strict) C -algebra is a V ∗ -2-functor X : C −→ Cat ( V ∗ ). Unpacking the definition in adjoint form, this consists of afunction that assigns a V ∗ -category X ( c ) to each object c of C , together with action V ∗ -functors θ : C ( c, d ) ∧ X ( c ) −→ X ( d )such that the unit and composition diagrams of V ∗ -functors( ∗ ∐ ∗ ) ∧ X ( c ) id c ∧ id (cid:15) (cid:15) ∼ = / / X C ( c, c ) ∧ X ( c ) θ sssssssssss and C ( d, e ) ∧ C ( c, d ) ∧ X ( c ) id ∧ θ / / ◦∧ id (cid:15) (cid:15) C ( d, e ) ∧ X ( d ) θ (cid:15) (cid:15) C ( c, e ) ∧ X ( c ) θ / / X ( e )commute. Remark 4.19.
The unit diagram can equally well be expressed as a diagram of V -functors, with source ∗ × X ( c ).We do not discuss general C -pseudoalgebras here, leaving such considerationfor [14]. However, we will need a version of pseudoalgebras in the special case of C = F G starting in Section 7. We define these now. ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 27 Definition 4.20.
Let C be a V ∗ -2-category. A weak C -pseudoalgebra X is a V ∗ -pseudofunctor C / / /o/o/o Cat ( V ∗ ) . Unpacking the definition in adjoint form, this con-sists of a function that assigns a V ∗ -category X ( c ) to each c ∈ C , together with action V ∗ -functors θ : C ( c, d ) ∧ X ( c ) −→ X ( d )and invertible V ∗ -transformations C ( d, e ) ∧ C ( c, d ) ∧ X ( c ) id ∧ θ / / ◦∧ id (cid:15) (cid:15) ✓✓✓✓ (cid:5) (cid:13) ϕ C ( d, e ) ∧ X ( d ) θ (cid:15) (cid:15) C ( c, e ) ∧ X ( c ) θ / / X ( e )which are the identity when either morphism is the identity 1-cell. Moreover, the ϕ are required to be coherent as in Definition 4.5. Remark 4.21.
The strictness with respect to basepoints encoded in the definitionsabove expresses the intuition that additive zero objects should behave strictly inmultiplicative structures. The strictness with respect to identity arrows expressesthe intuition that identity operations should be the identity.
Remark 4.22.
Though it is not the choice made in this article, it is also possible toconsider less general C -pseudoalgebras. As Cat ( V ∗ ) is closed, these can be describedas V ∗ -pseudofunctors C / / /o/o/o Cat ( V ∗ ) that are strictly functorial on a subcategory B ⊂ C . Different choices of B lead to several possible versions of pseudoalgebras.While we will only deal with the weakest possible variant of pseudoalgebras here,the different types of pseudomorphisms described in the next definition will play animportant role. We will return to consider pseudoalgebras in more detail in [14]. Definition 4.23.
Let C be a V ∗ -2-category and B ⊂ C a subcategory, and let X and Y be C -algebras (or weak C -pseudoalgebras). A ( C , B ) -pseudomorphism , F : X / / /o/o/o Y is a V ∗ -pseudotransformation between V ∗ -2-functors (or V ∗ -pseudo-functors) that is strict when restricted to B . Unpacking this definition in adjointform, this consists of V ∗ -functors F ( c ) : X ( c ) −→ Y ( c )together with invertible V ∗ -transformations δ as in the diagram(4.24) C ( c, d ) ∧ X ( c ) id ∧ F ( c ) / / θ (cid:15) (cid:15) ✑✑✑✑ (cid:4) (cid:12) δ C ( c, d ) ∧ Y ( c ) θ (cid:15) (cid:15) X ( d ) F ( d ) / / Y ( d )that are the identity transformation on the subcategory B ( c, d ) ∧ X ( c ). The δ mustsatisfy the relevant equalities of pasting diagrams relating to composition in C . If δ is always the identity, then F is a (strict) C -map . Thus the condition on B says that F restricts to a strict B -algebra map. We refer to the extreme case, inwhich B consists only of identity morphisms and basepoint morphisms, as weak C -pseudomorphisms .The following notion will also be needed later. Definition 4.25. A C -pseudomorphism F : X / / /o/o/o Y of C -pseudoalgebras is saidto be a level equivalence if each component F ( c ) : X ( c ) −→ Y ( c ) is an (internal)equivalence of V ∗ -categories. Definition 4.26. A C -transformation ω between C -pseudomorphisms E and F consists of V ∗ -transformations ω c : E ( c ) = ⇒ F ( c ) that are suitably compatible withthe V ∗ -transformations δ E and δ F , as in [10, Section 3.10]. We do not require the ω c to be isomorphisms.This definition is just a translation of the notion of a V ∗ -modification between V ∗ -pseudotransformations. We will only use the explicit description just given.These notions assemble to form various 2-categories of interest to us. Notation 4.27.
For a given V ∗ -2-category C with a subcategory B ⊂ C , we definethe following 2-categories, with C -transformations as the 2-cells in all cases. • C - Alg of C -algebras and strict C -maps; • C - Alg ps B of C -algebras and ( C , B )-pseudomorphisms; • C - PsAlg of weak C -pseudoalgebras and weak C -pseudomorphisms.We have inclusions C - Alg ⊂ C - Alg ps B ⊂ C - PsAlg . This article largely focuses on the 2-categories C - Alg ps B in the case that C isa category of operators. As noted in Notations 5.14 and 6.17, we will then fix therelevant B and drop the subscript B from the notation. However, pulling backalong the section ζ G in Section 7 will land in a category of type C - PsAlg , and thestrictification theorem in Section 8 will land in a category of type C - Alg .We end this subsection with two constructions on (pseudo)algebras.
Notation 4.28.
Given a V ∗ -pseudofunctor ξ : D / / /o/o/o C and a C -pseudoalgebra X ,we define the D -pseudoalgebra ξ ∗ X as the composite D ξ / / /o/o/o C X / / /o/o/o Cat ( V ∗ ) . This construction extends to C -pseudomorphisms and C -transformations, giving a2-functor ξ ∗ : C - PsAlg −→ D - PsAlg . If ξ is a strict V ∗ -2-functor, this construction restricts to give a 2-functor ξ ∗ : C - Alg −→ D - Alg . The 2-functor ξ ∗ preserves level equivalences of pseudoalgebras as defined in Definition 4.25. Definition 4.29.
Suppose given V ∗ -2-categories C and D . Given a C -pseudoalge-bra X and an D -pseudoalgebra Y , we define their external smash product X ∧ Y tobe the C ∧ D -pseudoalgebra given by the composite C ∧ D X ∧Y / / /o/o/o Cat ( V ∗ ) ∧ Cat ( V ∗ ) ∧ / / Cat ( V ∗ ) . If X and Y are strict algebras, so is X ∧ Y . ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 29 This construction extends to pseudomorphisms and transformations.4.3.
Permutative structures on V ∗ - -categories. In order to encode multiplica-tive structures on algebras, we use monoidal structures on V ∗ -2-categories, as de-fined in this section. Even in the case when V = Set , what we present here is notthe most general definition of a symmetric monoidal structure on a 2-category (see[3, 17]). Here, we present a rather strict notion in which the monoidal product isallowed to be a pseudofunctor, but must strictly satisfy associativity and unitality;while the symmetry is allowed to be a pseudotransformation, it must satisfy thesymmetry axiom strictly.
Definition 4.30. A strict pseudo-monoidal V ∗ -2-category consists of a V ∗ -2-category C together with an object I , and a V ∗ -pseudofunctor ⊛ : C ∧ C / / /o/o/o C that is strictly associative and strictly unital with respect to I . A pseudo-permutative V ∗ -2-category is a strict pseudo-monoidal V ∗ -2-category together with a V ∗ -pseudo-transformation C ∧ C ⊛ " " "b"b"b"b"b t / / C ∧ C ⊛ | | |< |< |< |< |< C ❴❴❴❴ + τ such that the following three axioms hold.(i) The following pasting diagram is equal to the identity of ⊛ : C ∧ C ⊛ ' ' 'g'g'g'g'g'g'g'g t / / C ∧ C ⊛ (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O t / / C ∧ C ⊛ w w w7 w7 w7 w7 w7 w7 w7 w7 C . ❴❴❴❴ + τ ❴❴❴❴ + τ (ii) The 2-cell C ∧ ∧ t / / C ∧ ⊛ ∧ id " " "b"b"b"b"b C ∧ t ∧ id < < ③③③③③③③③ id ∧ ⊛ / / /o/o/o/o/o/o/o/o/o ⊛ ∧ id " " "b"b"b"b"b C ∧ t / / ⊛ " " "b"b"b"b"b C ∧ ⊛ | | |< |< |< |< |< C ∧ ⊛ / / /o/o/o/o/o/o/o/o/o/o C ❴❴❴❴ + τ is equal to the 2-cell C ∧ ∧ t / / ⊛ ∧ id (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O id ∧ ⊛ " " "b"b"b"b"b C ∧ ⊛ ∧ id " " "b"b"b"b"b id ∧ ⊛ | | |< |< |< |< |< C ∧ t ∧ id < < ③③③③③③③③ ⊛ ∧ id " " "b"b"b"b"b C ∧ ⊛ " " "b"b"b"b"b ❴❴❴❴ + id ∧ τ C ∧ ⊛ | | |< |< |< |< |< C ∧ ⊛ / / /o/o/o/o/o/o/o/o/o/o ☎☎☎☎ > F τ ∧ id C . The unlabeled regions in both diagrams commute, the quadrilaterals by thestrict associativity of ⊛ and the pentagon by the naturality of t .If ⊛ is a strict V ∗ -2-functor and τ is a strict V ∗ -transformation, we say ( C , I, ⊛ , τ )is a permutative V ∗ -2-category . Remark 4.31.
Classically, a permutative category is a symmetric strict monoidalcategory, strict meaning that the product is strictly associative and unital. The def-inition above is similar, just done in the context of the 2-category of V ∗ -2-categories, V ∗ -pseudofunctors, and V ∗ -pseudotransformations. Thus “strict pseudo-monoidal”here means that ⊛ is a strictly associative and unital operation given by a V -pseudofunctor; it respects composition only up to coherent isomorphisms. Thestandard coherence theorem for permutative categories still applies in this case: forany permutation σ ∈ Σ k , there exists a unique composite of instances of τ that fitsin the diagram below. C ∧ k ⊛ k ` ` ` ` ` t σ / / C ∧ k ⊛ k ~ ~ ~> ~> ~> ~> ~> C ❴❴❴❴ + τ σ Here ⊛ k denotes the k -ary product induced by iterating ⊛ , and the map t σ sendsa k -tuple ( c , . . . , c k ) to ( c σ − (1) , . . . , c σ − ( k ) ). The 1-cell component of τ σ is the(unique) composite of instances of the 1-cell of τ that reorders c ⊛ · · · ⊛ c k −→ c σ − (1) ⊛ · · · ⊛ c σ − ( k ) . Definition 4.32.
Let ( C , I, ⊛ , τ ) and ( D , I ′ , ⊛ ′ , τ ′ ) be pseudo-permutative V ∗ -2-categories. A symmetric monoidal pseudofunctor (Ψ , µ ) : C / / /o/o/o D consists of a V ∗ -pseudofunctor Ψ : C / / /o/o/o D such that Ψ( I ) = I ′ , together with a V ∗ - pseudo-transformation C ∧ C Ψ ∧ Ψ / / /o/o/o ⊛ (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O ✡✡✡✡ (cid:1) (cid:9) µ D ∧ D ⊛ ′ (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O C Ψ / / /o/o/o/o/o D such that the following axioms hold.(i) µ is unital, meaning that its restrictions to { I } ∧ C and C ∧ { I } are theidentity transformation, where { I } ⊂ C denotes the discrete V ∗ -2-category onthe single object I . (ii) µ is associative, meaning that C ∧ C ∧ C Ψ ∧ Ψ ∧ Ψ / / /o/o/o/o/o ⊛ ∧ id (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O ✒✒✒✒ (cid:5) (cid:13) µ ∧ id D ∧ D ∧ D ⊛ ′ ∧ id (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O C ∧ C Ψ ∧ Ψ / / /o/o/o/o/o/o/o/o ⊛ (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O ✒✒✒✒ (cid:5) (cid:13) µ D ∧ D ⊛ ′ (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O C Ψ / / /o/o/o/o/o/o/o/o/o/o D = C ∧ C ∧ C Ψ ∧ Ψ ∧ Ψ / / /o/o/o/o/o id ∧ ⊛ (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O ✒✒✒✒ (cid:5) (cid:13) id ∧ µ D ∧ D ∧ D id ∧ ⊛ ′ (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O C ∧ C Ψ ∧ Ψ / / /o/o/o/o/o/o/o/o ⊛ (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O ✒✒✒✒ (cid:5) (cid:13) µ D ∧ D ⊛ ′ (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O C Ψ / / /o/o/o/o/o/o/o/o/o/o D . The V ∗ -object of morphisms in { I } is V { I , id I } ∼ = ∗ ∐ ∗ . Thus, { I } ∧ C ∼ = C . ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 31 The vertical boundaries of these diagrams are equal because of the associativ-ity of ⊛ and ⊛ ′ .(iii) The following equality of pasting diagrams holds: C ∧ C Ψ ∧ Ψ / / /o/o/o ⊛ (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O ⑤⑤⑤⑤ z (cid:2) µ D ∧ D ⊛ ′ (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O t & & ◆◆◆◆◆◆◆◆ ❴❴❴❴ k s τ ′ D ∧ D ⊛ ′ x x x8 x8 x8 x8 x8 x8 C Ψ / / /o/o/o/o/o D = C ∧ C ⊛ (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O t & & ▼▼▼▼▼▼▼▼ ❴❴❴❴ k s τ Ψ ∧ Ψ / / /o/o/o D ∧ D t & & ◆◆◆◆◆◆◆◆ C ∧ C ⊛ x x x8 x8 x8 x8 x8 x8 Ψ ∧ Ψ / / /o/o/o ❴❴❴❴ k s µ D ∧ D ⊛ ′ x x x8 x8 x8 x8 x8 x8 C Ψ / / /o/o/o/o/o D . If Ψ is a strict V ∗ -2-functor, we refer to (Ψ , µ ) as a symmetric monoidal 2-functor.5. The multicategory of D -algebras and the multifunctor R Nonequivariantly, categories of operators were introduced on the space level inorder to mediate the passage from algebras over an E ∞ -operad in spaces to special F -spaces when comparing the operadic and Segalic infinite loop space machines[37]. Algebras over a category of operators are a generalization of both F -spaces(aka Γ-spaces) and algebras over an E ∞ -operad. In this section we discuss theircategorical analogues. Equivariant categories of operators were studied in [46, 39],and their categorical analogues will be introduced in Section 6.We show that for a category of operators D with pseudo-commutative structure,as defined in Section 5.2, there is a multicategory of algebras over D . For categoriesof operators coming from operads, the necessary structure arises from a pseudo-commutative structure on the operad, as in Definition 3.10. We make all of thisprecise in this section.5.1. Categories of operators over F . The definitions in this subsection arecategorical analogues of definitions in [37]. We give the definitions in the setting of V -2-categories. Definition 5.1.
Recall that F denotes the category of based sets n = { , , . . . , n } with basepoint 0 and let Π denote its subcategory of morphisms φ : m −→ n suchthat | φ − ( j ) | = 0 or 1 for 1 ≤ j ≤ n . We often use the abbreviated notation φ j = | φ − ( j ) | . We regard F and Π as discrete 2-categories, meaning that they haveonly identity 2-cells. Via Remark 4.13, we then regard them as V ∗ -2-categories. Definition 5.2. A Cat ( V ) -category of operators D over F , abbreviated Cat ( V )- CO over F , is a V -2-category whose objects are the based sets n for n ≥ V -2-functors Π ι / / D ξ / / F such that ι and ξ are the identity on objects and ξ ◦ ι is the inclusion. A morphism ν : D −→ E of Cat ( V )- CO s over F is a V -2-functor over F and under Π. Definition 5.3. A Cat ( V )- CO D over F is reduced if is a zero object, andwe then say that D is a Cat ( V ∗ ) -category of operators over F . We shall restrictattention to Cat ( V ∗ )-categories of operators over F . Remark 5.4.
By Propositions 4.3 and 4.4, if D is a Cat ( V ∗ )- CO over F , then D is a V ∗ -2-category and ι and ξ are V ∗ -2-functors; that is, D is a V ∗ -2-categoryover F and under Π. A morphism D −→ E of reduced Cat ( V )- CO s over F isnecessarily reduced since it must send to ; thus it is a V ∗ -2-functor over F andunder Π.Let O be an operad in Cat ( V ). We can associate to it a category of operators D = D ( O ) over F by letting D ( m , n ) = a φ ∈ F ( m , n ) Y ≤ j ≤ n O ( φ j ) . Composition is induced from the structural maps γ of O . To write formulas in-stead of diagrams, we use elementwise notation, writing c i ∈ O ( φ j ) for objects andmorphisms in O ( φ j ). For ( φ, c , . . . , c n ) : m −→ n and ( ψ, d , . . . , d p ) : n −→ p ,define(5.5) ( ψ, d , . . . , d p ) ◦ ( φ, c , . . . , c n ) = (cid:16) ψ ◦ φ, Y ≤ j ≤ p γ ( d j ; Y ψ ( i )= j c i ) ρ j ( ψ, φ ) (cid:17) . The c i with ψ ( i ) = j are ordered by the natural order on their indices i , and ρ j ( ψ, φ )is that permutation of | ( ψ ◦ φ ) − ( j ) | letters which converts the natural ordering of( ψ ◦ φ ) − ( j ) as a subset of { , . . . , m } to its ordering obtained by regarding it as ` ψ ( i )= j φ − ( i ), so ordered that elements of φ − ( i ) precede elements of φ − ( i ′ ) if i < i ′ and each φ − ( i ) has its natural ordering as a subset of { , . . . , m } . Whenit is clear which φ and ψ are being composed, we abbreviate the notation for thepermutation ρ j ( ψ, φ ) to ρ j . Proposition 5.6. [37, Construction 4.1]
The above specification makes D ( O ) intoa category of operators over F , and it is reduced if O is reduced.Proof. The map ξ : D −→ F sends ( φ, c , . . . , c n ) to φ . Recall that any morphism φ in Π satisfies φ j ≤ j >
0. The inclusion ι : Π −→ D sends φ : m −→ n to( φ, c , . . . , c n ), where c j = ∈ O (1) if φ j = 1 and c j = ∗ ∈ O (0) if φ j = 0. (cid:3) Pseudo-commutative categories of operators over F . In analogy withour definition of pseudo-commutativity of an operad, we define a compatible notionof pseudo-commutativity of a category of operators D over F . The categories Π and F are permutative under the smash product of finite based sets, as we now recall.On objects, m ∧ p is defined to be mp . Given φ ∈ F ( m , n ) and ψ ∈ F ( p , q ) theirsmash product φ ∧ ψ ∈ F ( mp , nq ) is defined, in parallel to (3.6), as the restrictionof the map φ × ψ : m × p −→ n × q along the (based) inclusion mp ֒ → m × p that is given by (3.5) away from thebasepoint. The symmetry isomorphisms τ are given by the permutations τ m,p of Definition 3.8 which reorder the sets mp from lexicographic to reversed lexico-graphic ordering. The inclusion of Σ in F identifies ⊗ with ∧ . We will continue touse the symbol ⊗ for emphasis when dealing with permutations.Recall the notion of pseudo-permutative V ∗ -2-category from Definition 4.30. ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 33 Definition 5.7. A pseudo-commutative structure on D is a pseudo-permutativestructure ( D , , ⊛ , τ ) such that(1) ⊛ restricts to ∧ on Π ∧ Π and projects to ∧ on F (in the sense that ξ ◦ ⊛ = ∧ ◦ ( ξ ∧ ξ ));(2) ⊛ restricts to a strict V ∗ -2-functor on Π ∧ D and D ∧ Π;(3) τ restricts to the symmetry on Π given in Definition 3.8.We identify the pieces of this definition explicitly. First note that condition (1)implies in particular that ⊛ = ∧ on objects. The fact that ⊛ is a V ∗ -pseudofunctormeans that there is a collection of invertible V ∗ -transformations(5.8) D ( n , p ) ∧ D ( r , s ) ∧ D ( m , n ) ∧ D ( q , r ) ⊛ ∧ ⊛ / / id ∧ t ∧ id ∼ = (cid:15) (cid:15) ✓✓✓✓ (cid:5) (cid:13) ϑ D ( n ∧ r , p ∧ s ) ∧ D ( m ∧ q , n ∧ r ) ◦ (cid:15) (cid:15) D ( n , p ) ∧ D ( m , n ) ∧ D ( r , s ) ∧ D ( q , r ) ◦∧◦ (cid:15) (cid:15) D ( m , p ) ∧ D ( q , s ) ⊛ / / D ( m ∧ q , p ∧ s )relating ⊛ to composition.Condition (2), which is necessary for Theorem 6.14, translates to requiring ϑ to be the identity (so that the diagram commutes) when either both D ( n , p ) and D ( r , s ) are restricted to Π or both D ( m , n ) and D ( q , r ) are restricted to Π.Writing this out elementwise, on 1-cells it means that, whenever the compositesare defined, ( c ⊛ d ) ◦ ( a ∧ b ) = ( c ◦ a ) ⊛ ( d ◦ b )and ( a ∧ b ) ◦ ( c ⊛ d ) = ( a ◦ c ) ⊛ ( b ◦ d )where c and d are morphisms of D and a and b are morphisms of Π. We thinkof this as saying that the monoidal structure on ⊛ is strict relative to Π. Asthe pseudofunctoriality constraint for ⊛ , the V ∗ -transformations ϑ must satisfy acondition with respect to triple composition. The condition on ⊛ being strictlyassociative imposes another set of conditions on ϑ .Condition (3) means that the 1-cell constraint of τ at the object ( m , p ) is the per-mutation τ m,p thought of as a morphism in D , and the pseudonaturality constraintis an invertible V ∗ -transformation(5.9) D ( m , n ) ∧ D ( p , q ) ⊛ ◦ t / / ⊛ (cid:15) (cid:15) ✒✒✒✒ (cid:5) (cid:13) ˆ τ D ( pm , qn ) ( τ m,p ) ∗ (cid:15) (cid:15) D ( mp , nq ) ( τ n,q ) ∗ / / D ( mp , qn ) that is the identity when restricted to the subcategory Π( m , n ) ∧ Π( p , q ). Thesepseudonaturality constraints must be compatible with composition in D and thepseudofunctoriality constraint ϑ . Remark 5.10.
For later use, we emphasize a particular consequence of the strict-ness relative to Π here. Let a ∈ Π( m ′ , m ) and b ∈ Π( q ′ , q ). The compatibility of ϑ with triple composition together with condition (2) implies that D ( n , p ) ∧ D ( r , s ) ∧ D ( m , n ) ∧ D ( q , r ) ⊛ ∧ ⊛ / / id ∧ id ∧ a ∗ ∧ b ∗ (cid:15) (cid:15) D ( n ∧ r , p ∧ s ) ∧ D ( m ∧ q , n ∧ r ) id ∧ ( a ∧ b ) ∗ (cid:15) (cid:15) D ( n , p ) ∧ D ( r , s ) ∧ D ( m ′ , n ) ∧ D ( q ′ , r ) ⊛ ∧ ⊛ / / id ∧ t ∧ id ∼ = (cid:15) (cid:15) ✓✓✓✓ (cid:5) (cid:13) ϑ D ( n ∧ r , p ∧ s ) ∧ D ( m ′ ∧ q ′ , n ∧ r ) ◦ (cid:15) (cid:15) D ( n , p ) ∧ D ( m ′ , n ) ∧ D ( r , s ) ∧ D ( q ′ , r ) ◦∧◦ (cid:15) (cid:15) D ( m ′ , p ) ∧ D ( q ′ , s ) ⊛ / / D ( m ′ ∧ q ′ , p ∧ s )is equal to D ( n , p ) ∧ D ( r , s ) ∧ D ( m , n ) ∧ D ( q , r ) ⊛ ∧ ⊛ / / id ∧ t ∧ id ∼ = (cid:15) (cid:15) ✓✓✓✓ (cid:5) (cid:13) ϑ D ( n ∧ r , p ∧ s ) ∧ D ( m ∧ q , n ∧ r ) ◦ (cid:15) (cid:15) D ( n , p ) ∧ D ( m , n ) ∧ D ( r , s ) ∧ D ( q , r ) ◦∧◦ (cid:15) (cid:15) D ( m , p ) ∧ D ( q , s ) ⊛ / / a ∗ ∧ b ∗ (cid:15) (cid:15) D ( m ∧ q , p ∧ s ) ( a ∧ b ) ∗ (cid:15) (cid:15) D ( m ′ , p ) ∧ D ( q ′ , s ) ⊛ / / D ( m ′ ∧ q ′ , p ∧ s )The two unlabeled squares have instances of ϑ that are the identity because ofcondition (2). The boundaries on both diagrams are equal since composition in D is strict. This equality expresses a condition on ϑ for when the first terms of atriple composition come from Π. There are similar conditions for when the middleand the last terms come from Π. Definition 5.11.
We define a map (Ψ , µ ) : D −→ E of pseudo-commutative cat-egories of operators to be a symmetric monoidal 2-functor (Definition 4.32) suchthat Ψ is a map of Cat ( V ∗ )- CO s over F and the restriction of µ to the subcategoryΠ ∧ Π is the identity transformation.We defer the proof of the following theorem to Section 12. It ensures that our def-initions of pseudo-commutativity for operads and for their associated categories ofoperators are compatible. The verification is essentially combinatorial bookkeepingand is painstaking rather than hard.
ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 35 Theorem 5.12.
Let O be a pseudo-commutative operad in Cat ( V ) . Then D = D ( O ) is a pseudo-commutative category of operators. Remark 5.13.
The construction is functorial. With the appropriate definition of apseudo-commutative morphism
O −→ P of pseudo-commutative operads, the map D ( O ) −→ D ( P ) is pseudo-commutative. In analogy with Lemma 3.12, when O and P are chaotic, any morphism of operads between them is necessarily pseudo-commutative.5.3. The multicategory of D -algebras. For a category of operators D over F ,we consider D -algebras as defined in Definition 4.18. As indicated in Remark 4.22,there is a more general notion of D -pseudoalgebra, but we defer discussion ofthat to [14]. Recall that we have the notions of ( D , Π)-pseudomorphism and D -transformation from Definitions 4.23 and 4.26. Notation 5.14.
For a category of operators D over F , we denote by D - Alg ps the2-category of strict D -algebras, ( D , Π)-pseudomorphisms, and D -transformations.This 2-category was denoted by D - Alg ps Π in Notation 4.27, but we now fix B = Πand drop it from the notation.This notion, with its strictness with respect to Π, is essential for the constructionof P in the left column of (1.2), as we explain in Remark 6.3.Let D be a reduced pseudo-commutative category of operators over F . Wedefine the multicategory Mult ( D ) of D -algebras, which amounts to defining the k -ary morphisms. As said before, we set it up to have its objects be D -algebras,although with only slightly more work we could equally well have set it up to haveits objects be D -pseudoalgebras.Recall from Section 4.2 that a D -algebra is given by a V ∗ -2-functor X : D −→ Cat ( V ∗ ), which can be expressed in adjoint form as in Definition 4.18. Thus theaction of D on X is given by V ∗ -functors θ : D ( m , n ) ∧ X ( m ) −→ X ( n ) . Let D ∧ k denote the k -fold smash power. Following Definition 4.29, given D -algebras X , . . . , X k , we have the external smash product X ∧ . . . ∧X k . It sends anobject ( n , . . . , n k ) of D ∧ k to X ( n ) ∧ · · · ∧ X k ( n k ), with action map θ k given bythe composite V i D ( m i , n i ) ∧ V i X ( m i ) t ∼ = / / V i D ( m i , n i ) ∧ X ( m i ) V θ / / V i X ( n i ) , where the first map is the appropriate shuffle. For a D -algebra Y , we consider the D ∧ k -pseudoalgebra D ∧ k ⊛ k / / /o/o/o D Y / / Cat ( V ∗ ) . The conditions in Definition 5.7 imply that
Y ◦ ⊛ k restricts to a strict Π ∧ k -algebra.Since Π ∧ k is discrete, this is a functor from Π ∧ k to the underlying 1-category of Cat ( V ∗ ). Definition 5.15.
Let D be a reduced pseudo-commutative category of operatorsover F . We define a (symmetric) multicategory Mult ( D ) of D -algebras as follows.The objects are D -algebras. For objects X i , 1 ≤ i ≤ k , and Y , a k -ary morphism X −→ Y consists of a ( D ∧ k , Π ∧ k )-pseudomorphism F : X ∧ . . . ∧X k / / /o/o/o Y ◦ ⊛ k . Recall that this is the same as saying that F is a V ∗ -pseudotransformation D ∧ k X ∧···∧X k / / ⊛ k (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O ✏✏✏✏ (cid:4) (cid:12) F Cat ( V ∗ ) ∧ k ∧ k (cid:15) (cid:15) D Y / / Cat ( V ∗ )that is strict when restricted to Π ∧ k .Given a j i -ary morphism E i : ( X i, , . . . , X i,j i ) −→ Y i for i = 1 , . . . k , and a k -ary morphism F : ( Y , . . . , Y k ) −→ Z , the composite is defined by the pastingdiagram below, where the right hand 2-cell is the associativity isomorphism for ∧ on Cat ( V ∗ ).(5.16) D ∧ j X , ∧···∧X k,jk / / ⊛ j (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O V i ⊛ ji % % %e%e%e%e%e%e%e ✍✍✍✍ (cid:3) (cid:11) E ∧···∧ E k ❴❴❴❴ k s Cat ( V ∗ ) ∧ j V i ∧ ji v v ♠♠♠♠♠♠♠♠♠♠♠♠ ∧ j (cid:15) (cid:15) D ∧ k Y ∧···∧Y k / / ✎✎✎✎ (cid:3) (cid:11) F ⊛ k y y y9 y9 y9 y9 y9 y9 y9 Cat ( V ∗ ) ∧ k ∧ k ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ D Z / / Cat ( V ∗ ) . Finally, we specify the symmetric structure on the multicategory
Mult ( D ).Given a permutation σ ∈ Σ k and a k -ary morphism F : ( X , . . . , X k ) −→ Y , the k -ary morphism F σ : ( X σ (1) , . . . , X σ ( k ) ) −→ Y is defined by the pasting diagram D ∧ k ⊛ k (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O(cid:15)O t σ % % ❑❑❑❑❑❑❑❑❑❑❑ ❴❴❴❴ k s τ − σ X σ (1) ∧···∧X σ ( k ) / / ❴❴❴❴ k s t σ Cat ( V ∗ ) ∧ kt σ v v ♠♠♠♠♠♠♠♠♠♠♠♠ ∧ k (cid:15) (cid:15) D ∧ k X ∧···∧X k / / ✎✎✎✎ (cid:3) (cid:11) F ⊛ k y y y9 y9 y9 y9 y9 y9 y9 Cat ( V ∗ ) ∧ k ∧ k ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ D Y / / Cat ( V ∗ ) . Here the different maps called t σ send a k -tuple ( a , . . . , a k ) to ( a σ − (1) , . . . , a σ − ( k ) ),and τ σ is the invertible V ∗ -pseudotransformation of Remark 4.31.We now unpack this definition. In what follows, given a k -tuple ( n , . . . , n k ) ofnatural numbers, we write n = n · · · · · n k . ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 37 A k -ary morphism F = ( F, δ ) : ( X , . . . , X k ) −→ Y consists of V ∗ -functors F : X ( n ) ∧ · · · ∧ X k ( n k ) −→ Y ( n ) , together with invertible V ∗ -transformations δ in the following diagrams, in which1 ≤ i ≤ k .(5.17) V i D ( m i , n i ) ∧ V i X i ( m i ) id ∧ F / / t ∼ = (cid:15) (cid:15) ✍✍✍✍ (cid:3) (cid:11) δ V i D ( m i , n i ) ∧ Y ( m ) ⊛ k ∧ id (cid:15) (cid:15) V i D ( m i , n i ) ∧ X i ( m i ) V i θ (cid:15) (cid:15) D ( m , n ) ∧ Y ( m ) θ (cid:15) (cid:15) V i X i ( n i ) F / / Y ( n )We require δ to be the identity when restricted to Π ∧ k . The δ must satisfy coherencediagrams related to composition and identities in D ∧ k . The latter are subsumed inthe conditions on Π. We defer writing out the details of the required conditions forcomposition to Section 11.3.Unpacking the action of σ , the component of ( F, δ ) σ = ( F σ, δσ ) at an object m , . . . , m k is defined by the following commutative diagram.(5.18) X σ (1) ( m ) ∧ · · · ∧ X σ ( k ) ( m k ) F σ / / t σ (cid:15) (cid:15) Y ( m ) X ( m σ − (1) ) ∧ · · · ∧ X k ( m σ − ( k ) ) F / / Y ( m ) Y ( τ − σ ) O O The invertible V ∗ -transformation δσ is obtained by whiskering the δ of (5.17), butusing the pseudocommutativity of D . Precisely, we construct δσ by the followingpasting diagram, where we write υ for σ − . Here the inner hexagon is (5.17), andthe outer hexagon is the corresponding diagram for F σ . On D ( m , n ) we denote by c σ the pre- and postcomposition with τ σ : m = ^ i m i −→ ^ i m υ ( i ) and τ − σ : ^ i n υ ( i ) −→ ^ i n i = n , respectively. (5.19) V i D ( m i , n i ) ∧ V i X σ ( i ) ( m i ) id ∧ Fσ / / t σ ∧ t σ * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ t (cid:15) (cid:15) ☛☛☛☛ (cid:1) (cid:9) ˆ τ ∧ id V i D ( m i , n i ) ∧ Y ( m ) ⊛ ∧ id (cid:15) (cid:15) V i D ( m υ ( i ) , n υ ( i ) ) ∧ V i X i ( m υ ( i ) ) t (cid:15) (cid:15) id ∧ F / / ✏✏✏✏ (cid:4) (cid:12) δ V i D ( m υ ( i ) , n υ ( i ) ) ∧ Y ( V i m υ ( i ) ) ⊛ ∧ id (cid:15) (cid:15) t − σ ∧Y ( τ − σ ) ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ V i D ( m i , n i ) ∧ X σ ( i ) ( m i ) V i θ (cid:15) (cid:15) t σ / / V i D ( m υ ( i ) , n σ − ( i ) ) ∧ X i ( m υ ( i ) ) V i θ (cid:15) (cid:15) D ( V i m υ ( i ) , V i n υ ( i ) ) ∧ Y ( V i m υ ( i ) ) c σ ∧Y ( τ − σ ) / / θ (cid:15) (cid:15) D ( m , n ) ∧ Y ( m ) θ (cid:15) (cid:15) V i X i ( n υ ( i ) ) F / / Y ( V i n υ ( i ) ) Y ( τ − σ ) * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ V i X σ ( i ) ( n i ) Fσ / / t σ ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ Y ( n ) The top and bottom trapezoids commute by the definition of
F σ . The left twotrapezoids commute trivially. The bottom right trapezoid commutes since
Y ∈
Mult ( D ) is a (strict) D -algebra. The top right trapezoid is filled by an invertible V ∗ -transformation ˆ τ given by the pseudonaturality constraint of the appropriate(unique) composition of instances of the pseudocommutativity of D .Recall Notation 4.28. Theorem 5.20.
Let (Ψ , µ ) : D −→ E be a map of pseudo-commutative categoriesof operators (Definition 5.11). Then pulling back along Ψ induces a (symmetric)multifunctor Ψ ∗ : Mult ( E ) −→ Mult ( D ) . Proof.
Given a k -ary morphism F : ( X , . . . , X k ) −→ Y in Mult ( E ), the multimor-phism Ψ ∗ ( F ) is defined as the pasting E ∧ k Ψ ∧ k / / ⊛ k (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O ✌✌✌✌ (cid:2) (cid:10) µ k D ∧ k X ∧···∧X k / / ⊛ k (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O ✏✏✏✏ (cid:4) (cid:12) F Cat ( V ∗ ) ∧ k ∧ k (cid:15) (cid:15) E Ψ / / D Y / / Cat ( V ∗ ) , where the 2-cell µ k denotes an appropriate composite of instances of µ , which isunique by the associativity of µ . Compatibility with the identity, composition, andthe symmetric group action follows from the axioms in Definition 5.11. (cid:3) Definition of the functor R . Let O be a reduced operad in Cat ( V ) with as-sociated Cat ( V ∗ )- CO D over F . We define a 2-functor R : O - Alg ps −→ D - Alg ps with the property that for an O -algebra A , the resulting D -algebra is defined onobjects by n
7→ A n , and we show in Section 5.5 that R extends to a multifunctor.We have the 2-category Π- Alg of Π-algebras, Π-morphisms, and Π-transformations,and we have the evident 2-functor R : Cat ( V ∗ ) −→ Π- Alg that sends a V ∗ -category A to the Π-algebra Π −→ Cat ( V ∗ ) whose value at n is A n . The injections,projections, and permutations of Π are sent to basepoint inclusions, projections, ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 39 and permutations of the A n . For a V ∗ -functor F : A −→ B , R F has component F n : A n −→ B n at n .The notation R records that R is right adjoint to the 2-functor L that sends a Π-algebra to its first V ∗ -category, L ( X ) = X ( ) [34, § R extends toa 2-functor R : O - Alg ps −→ D - Alg ps . When starting operadically, it is convenientto use × instead of ∧ . For an O -algebra A in Cat ( V ), we give R A a D -algebrastructure via a V -functor θ : D ( m , n ) × A m −→ A n that is compatible with basepoints and therefore descends to a V ∗ -functor on thesmash product. Writing φ j = | φ − ( j ) | again, this V -functor can be expressed as acomposition ` φ : m → n (cid:16) Q ≤ j ≤ n O ( φ j ) (cid:17) × A m / / ` φ : m → n Q ≤ j ≤ n (cid:0) O ( φ j ) × A φ j (cid:1) θ / / A n . On each component φ : m −→ n , the first map reorders A m ∼ = A φ ×A φ ×· · ·×A φ n and projects away A φ , while the second map is the product of n algebra structuremaps θ ( φ j ) : O ( φ j ) × A φ j −→ A .We next define R on morphisms. Thus let ( F, ∂ ∗ ) : A / / /o/o/o B be a pseudomor-phism of O -algebras. We define a ( D , Π)-pseudomorphism R ( F, ∂ ∗ ) = ( R F, δ ) : R A / / /o/o/o R B . The required V ∗ -transformation D ( m , n ) ∧ A m id ∧ F m / / θ (cid:15) (cid:15) ✒✒✒✒ (cid:5) (cid:13) δ m , n D ( m , n ) ∧ B mθ (cid:15) (cid:15) A n F n / / B n is obtained by passage to smash products from a coproduct of whiskerings of Q ≤ j ≤ n (cid:0) O ( φ j ) × A φ j (cid:1) Q id × F φj / / Q θ (cid:15) (cid:15) ✕✕✕✕ (cid:6) (cid:14) Q ∂ Q ≤ j ≤ n (cid:0) O ( φ j ) × B φ j (cid:1) Q θ (cid:15) (cid:15) A n F n / / B n along the reordering morphisms (cid:16) Q ≤ j ≤ n O ( φ j ) (cid:17) × A m −→ Q ≤ j ≤ n (cid:0) O ( φ j ) × A φ j (cid:1) .For an O -transformation ω : E = ⇒ F , we define the component of the D -transformation R ω at n as ω n . We leave it to the reader to fill in the detailsof the proof of the following result. Proposition 5.21.
The above data specifies a -functor R : O - Alg ps −→ D - Alg ps . The proof that R is a symmetric multifunctor. Now let O be a reducedpseudo-commutative operad in Cat ( V ) with associated pseudo-commutative cate-gory of operators D . Theorem 5.22.
The -functor R : O - Alg ps −→ D - Alg ps extends to a symmetricmultifunctor Mult ( O ) −→ Mult ( D ) .Proof. Let (
F, δ i ) : ( A , . . . , A k ) −→ B , be a k -ary morphism in Mult ( O ). Here F is a V ∗ -functor A ∧· · ·∧A k −→ B and δ i is given by V -transformations δ i ( n ) as in(3.15). We must construct a k -ary morphism R ( F, δ i ) = ( R F, R δ ) : ( R A , . . . , R A k ) −→ R B as in Definition 5.15.Writing n = n · · · n k as before, the component R A ( n ) ∧ · · · ∧ R A k ( n k ) −→ R B ( n )of R F is A n ∧ · · · ∧ A n k k ℓ −→ ( A ∧ · · · ∧ A k ) n F n −−→ B n , where ℓ is a based version of the map defined in Definition 3.20, using lexicographicordering. Next, we specify the V ∗ -transformations R δ in the following specializationof diagram (5.17).(5.23) V i D ( m i , n i ) ∧ V i A im i id ∧ R F / / t ∼ = (cid:15) (cid:15) ☛☛☛☛ (cid:1) (cid:9) R δ V i D ( m i , n i ) ∧ B m ⊛ ∧ id (cid:15) (cid:15) V i D ( m i , n i ) ∧ A m i i V i θ (cid:15) (cid:15) D ( m , n ) ∧ B mθ (cid:15) (cid:15) V i A n i i R F / / B n . Before passage to smash products, these transformations are constructed as dis-joint unions of products of compositions of the δ i ( n )’s. To see this, consider, forinstance, the case in which k = 2 and n = n = . We restrict further to thecomponents O ( m ) ⊆ D ( m , ) and O ( m ) ⊆ D ( m , ) corresponding to the maps φ : m −→ and ψ : m −→ that send all non-basepoint elements to 1. Then the V ∗ -transformation R δ is obtained by passage to smash products from the 2-cell O ( m ) × O ( m ) × A m × A m ∼ = (cid:15) (cid:15) id × R F / / ✏✏✏✏ (cid:4) (cid:12) O ( m ) × O ( m ) × B m m ⊛ × id (cid:15) (cid:15) O ( m ) × A m × O ( m ) × A m θ × θ (cid:15) (cid:15) O ( m m ) × B m m θ (cid:15) (cid:15) A × A F / / B of (v), the axiom for commutation of cells δ i and δ j , in Section 11.2, where Definition 3.14is completed. Now consider the general case of k = 2, with arbitrary n and n . ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 41 For the component of D ( m , n ) × D ( m , n ) indexed by maps φ : m −→ n and ψ : m −→ n in F , the required 2-cell is of the form Q ≤ j ≤ n O ( φ j ) × Q ≤ k ≤ n O ( ψ k ) × A m × A m θ × Π θ ) ◦ t (cid:15) (cid:15) ⊛ × R F / / ✔✔✔✔ (cid:6) (cid:14) Q j,k O (( φ ∧ ψ ) ( j,k ) ) × B m m Π θ (cid:15) (cid:15) A n × A n R F / / B n n and is a product of 2-cells of the previous type. We note that R δ is the identitywhen D k is restricted to Π k , by axioms (i) and (iii) in Definition 3.14. When k = 1,the construction above recovers that of Section 5.4. Axiom (v) of Definition 3.14implies that R preserves composition.We prove that R is symmetric by a comparison of the definitions here with thoseof Section 3.3. Remembering the lexicographic reordering, it is straightforwardto check by comparison of (3.16) with (5.18) that R ( F σ ) = ( R F ) σ for a k -arymorphism ( F, δ i ) of Mult ( O ). The equality of pasting diagrams required to ensurethat the V ∗ -transformations in ( R ( F, δ i )) σ and R (( F, δ i ) σ ) are equal follows fromaxiom (v) of Definition 3.14. (cid:3) The multicategory of D G -algebras and the multifunctor P We introduced
Cat ( V )-categories of operators in Section 5, as well as the multi-category Mult ( D ) associated to any pseudo-commutative category of operators D .In this section, we finally bring in equivariance, starting in Section 6.1, where wespecialize the content of the previous sections to the category G V of G -objects in V . In Section 6.2 we introduce Cat ( G V )-categories of operators over F G , the cate-gory of finite based G -sets. A key idea here is Construction 6.7, which allows us toprolong from equivariant categories of operators over F to equivariant categories ofoperators over F G . We introduce the multicategory Mult ( D G ) in Section 6.5 andextend the prolongation functor to a symmetric multifunctor in Section 6.6. Muchof this section is precisely parallel to the previous one.6.1. G V -categories and G V ∗ -categories. So far, equivariance has not enteredinto the picture and yet everything we have done applies equally well equivariantly,as we now explain. Start again with a category V satisfying Assumptions 2.2 and2.3, such as the category U of spaces, and let G be a finite group. An action of G onan object X of V can be specified in several equivalent ways. One is to regard G asa group in V via Remark 4.13 and to require a map G × X −→ X in V that satisfiesthe evident unit and associativity properties, expressed diagrammatically. Anotheris to regard G as a category with one object and to require a functor G −→ V thatsends the one object to X . We have the evident notion of a G -map X −→ Y . Let G V denote the category of G -objects in V and G -maps between them. Then G V is bicomplete, with limits and colimits created in V and given the induced ac-tions by G . With the second description, this is a standard fact about functor cat-egories. Therefore G V satisfies Assumption 2.2. Similarly, we have the 2-category Cat ( G V ) of categories internal to G V , which can be identified with the 2-categoryof G -objects in Cat ( V ). Hence G V satisfies Assumption 2.3 as well. Thus we canreplace V by G V and everything we have said so far applies verbatim. Remark 6.1.
As just noted, we can think of a G V -category as a V -category C together with V -functors g : C −→ C for g ∈ G . Then a G V -functor F : B −→ C isa V -functor F such that the diagrams B g / / F (cid:15) (cid:15) B F (cid:15) (cid:15) C g / / C commute. A G V -transformation ν : E = ⇒ F is a V -transformation such that ν : Ob B −→ Mor C is G -equivariant. This is equivalent to having the followingequality of pasting diagrams. B g / / E (cid:6) (cid:6) F (cid:24) (cid:24) ❴❴❴❴ k s ν B E (cid:15) (cid:15) C g / / C = B g / / F (cid:15) (cid:15) B E (cid:6) (cid:6) F (cid:24) (cid:24) ❴❴❴❴ k s ν C g / / C The terminal object of V , with trivial G -action, is terminal in G V , and we havethe category G V ∗ of based objects in G V , which can be identified with the categoryof G -objects in V ∗ . We also have the 2-category Cat ( G V ∗ ) of categories internalto G V ∗ , which can be identified with the 2-category of G -objects in Cat ( V ∗ ); ananalogue of Remark 6.1 applies in this case as well.We also have G V -2-categories, which are defined to be categories enriched in Cat ( G V ), using cartesian products, and G V ∗ -2-categories, which are categoriesenriched in Cat ( G V ∗ ), defined using smash products. We emphasize that G doesnot act on the collection of objects of a G V -2-category. It acts on the V -categoriesof morphisms.What changes is that we now build finite G -sets into the picture. We workwith categories of operators D G over F G , our chosen permutative model of thecategory of based finite G -sets. We define these in Section 6.2, we define pseudo-commutativity for them in Section 6.3, and we define algebras and pseudoalgebrasover them in Section 6.4. When D G is pseudo-commutative, we define a multicate-gory with underlying category D G - Alg ps in Section 6.5. We define the prolongationfunctor P from D -algebras to D G -algebras and show that it extends to a symmetricmultifunctor P : Mult ( D ) −→ Mult ( D G ) in Section 6.6.6.2. Categories of operators over F G . The definition of a category of operatorsover F G in this section is parallel to that of a category of operators over F inSection 5.1, and it is the categorical analogue of the definition given in [39] on the ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 43 space level. After giving the relevant definitions, we show how we can go back andforth between categories of operators over F and categories of operators over F G . Definition 6.2.
Let F G be the following model of the G -category of based finite G -sets. An object in F G consists of a based set n together with a G -action prescribedby a homomorphism α : G −→ Σ n . We denote this object by n α .The morphismsare defined to be all based functions, not just the equivariant ones, and we let G actby conjugation on the set of morphisms. Thus F G can be viewed as a G -categorywhere G acts trivially on objects.Let Π G ⊂ F G be the sub G -category of morphisms φ : m α −→ n β such that φ j = 0 or 1 for 1 ≤ j ≤ n . Write n for n ε , where ε is the trivial homomorphism.That fixes compatible embeddings of Π in Π G and F in F G . Note that is a zeroobject in Π G and F G . Remark 6.3.
We think of Σ n as the subset of isomorphisms in Π( n , n ). For abased G -set n α , the homomorphism α thus maps G to Π( n , n ), which is containedin all categories of operators of either type. We have built strictness with respectto Π into all of our structures, and the strictness with respect to permutations iscrucial in dealing with equivariance, in particular in constructing the prolongationfunctor P . Definition 6.4. A Cat ( G V ) -category of operators D G over F G , which we abbre-viate to Cat ( G V )- CO over F G , is a G V -2-category whose objects are the based G -sets n α for n ≥
0, together with G V -2-functorsΠ G ι G / / D G ξ G / / F G such that ι G and ξ G are the identity on objects and ξ G ◦ ι G is the inclusion. Amorphism ν G : D G −→ E G of Cat ( G V )- CO s over F G is a G V -2-functor over F G and under Π G .For a Cat ( G V )- CO D G over F G , let D denote the full subcategory on theobjects n = n ε with trivial G -action. This is the underlying Cat ( G V )- CO over F of D G . Definition 6.5. A Cat ( G V ) -category of operators D G is reduced if is a zeroobject, and we then say that D is a Cat ( G V ∗ ) -category of operators over F G . Weshall restrict attention to Cat ( G V ∗ )-categories of operators over F G . Remark 6.6.
As in Remark 5.4, a
Cat ( G V ∗ )- CO D G over F G is a G V ∗ -2-category,with ι G and ξ G G V ∗ -2-functors, and a morphism of reduced Cat ( G V )- CO s over F G is reduced and is thus a G V ∗ -2-functor over F G and under Π G . Construction 6.7.
We construct a prolongation functor P from the category of Cat ( G V ∗ )- CO s over F to the category of Cat ( G V ∗ )- CO s over F G . Let D be a Cat ( G V ∗ )- CO over F . Define the morphism G V ∗ -category P D ( m α , n β ) to be acopy of D ( m , n ), but with G -action induced by conjugation and the original given G -action on D ( m , n ). Explicitly, the action of g ∈ G on P D ( m α , n β ), which weshall call P g when α and β are understood, is the composite(6.8) P g := D ( m , n ) α ( g − ) ∗ / / D ( m , n ) g / / D ( m , n ) β ( g ) ∗ / / D ( m , n ) . Here α ( g − ) ∗ and β ( g ) ∗ are defined by precomposition with α ( g − ) and postcom-position with β ( g ); we think of them as prewhiskerings and postwhiskerings. Com-position is inherited from D and is equivariant. Observing that Π G and F G are theprolongations of Π and F , the inclusion ι G and projection ξ G are inherited from D as P ι and P ε , This uses the functoriality of P , which we now explain. For a mapof Cat ( G V ∗ )- CO s ν : D → E , we define P ν : P D ( m α , n β ) −→ P E ( m α , n β )to just be ν ; it is equivariant with respect to the new action because ν is a G V ∗ -2-functor and thus is compatible with the G -action and with precomposition andpostcomposition with maps in Π. Proposition 6.9. If D G is a Cat ( G V ∗ ) - CO over F G , then D G ∼ = P D , where D is the underlying Cat ( G V ∗ ) - CO over F of D G .Proof. Let D G be a Cat ( G V ∗ )- CO over F G . Let id α ∈ Π G ( m , m α ) and id α ∈ Π G ( m α , m ) be the morphisms given by the identity map on the set m . They arenot identity morphisms but rather are mutual inverses in Π G , and hence in D G .Since G acts by conjugation on Π G ( m , m α ) and Π G ( m α , m ), the action of g sendsid α and id α to the maps on m given by α ( g ) and α ( g ) − , respectively.Precomposition with id α and postcomposition with id β induce an isomorphismof V ∗ -categories(6.10) D G ( m α , n β ) −→ D G ( m , n ) = D ( m , n ) . The above observations and the fact that composition in D G is G -equivariantimply that this map becomes G -equivariant when we endow the target with theaction defined on P D ( m α , n β ), giving the desired isomorphism D G ∼ = P D . (cid:3) Remark 6.11.
The map of V ∗ -categories in (6.10) induces a V ∗ -2-functor D G −→ D . It is an inverse up to invertible V ∗ -2-natural transformation to the inclusion D −→ D G . Thus D and D G are equivalent as V ∗ -2-categories, but not as G V ∗ -2-categories. Definition 6.12.
For a reduced operad O in Cat ( G V ), define the associated cat-egory of operators D G ( O ) over F G to be the prolongation P ( D ( O )).A more explicit description is given on the space level in [39, Definition 5.1]. Pseudo-commutative categories of operators over F G . Observe thatΠ G and F G are permutative under the smash product of finite based G -sets. Onunderlying sets, the smash product and its symmetry isomorphism are defined justas for Π and F . Recall that when we restrict to Σ, we denote the smash product ⊗ . This can be thought of as a collection of maps Σ m × Σ n −→ Σ mn . Thenhomomorphisms α : G −→ Σ m and β : G −→ Σ n have the product homomorphism α ⊗ β given by applying ⊗ elementwise; that is, ( α ⊗ β )( g ) = α ( g ) ⊗ β ( g ). Numbered references to [39] refer to the first version posted on ArXiv; they will be changedwhen the published version appears.
ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 45 Then m α ∧ n β = mn α ⊗ β , and the smash product F G ( m α , p γ ) ∧ F G ( n β , q δ ) −→ F G ( mn α ⊗ β , pq γ ⊗ δ )is G -equivariant.The following definition is precisely analogous to Definition 5.7. Definition 6.13. A pseudo-commutative structure on a category of operators D G over F G is a pseudo-permutative structure ( D G , , ⊛ , τ ) such that(1) ⊛ restricts to ∧ on Π G ∧ Π G and projects to ∧ on F G (in the sense that ξ G ◦ ⊛ = ∧ ◦ ( ξ G ∧ ξ G ));(2) ⊛ restricts to a strict G V ∗ -2-functor on Π G ∧ D G and D G ∧ Π G ;(3) τ restricts to the symmetry on Π G .Define a map of pseudo-commutative categories of operators over F G as in Definition 5.11.The following theorem states that the prolongation of a pseudo-commutativecategory of operators is again pseudo-commutative. Theorem 6.14.
Let D be a pseudo-commutative Cat ( G V ∗ ) -category of operatorsover F . Then P D is a pseudo-commutative Cat ( G V ∗ ) -category of operators over F G and the inclusion ( D , Π) −→ ( P D , Π G ) preserves the pseudo-commutativestructure.Proof. We define the G V ∗ -pseudofunctor ⊛ : P D ∧ P D / / /o/o/o P D as follows. Onobjects, ⊛ is just ∧ ; that is, m α ⊛ n β = mn α ⊗ β . On G V ∗ -categories of morphisms, ⊛ : P D ( m α , p γ ) ∧ P D ( n β , q δ ) −→ P D ( mn α ⊗ β , pq γ ⊗ δ )is just ⊛ : D ( m , p ) ∧ D ( n , q ) −→ D ( mn , pq ) . We need to show that ⊛ is equivariant with respect to the action of G on P D (seeRemark 6.1). The equivariance is encoded in the following commutative diagram.(6.15) D ( m , p ) ∧ D ( n , q ) ⊛ (cid:15) (cid:15) α ( g − ) ∗ ∧ β ( g − ) ∗ ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ P g ∧ P g / / D ( m , p ) ∧ D ( n , q ) ⊛ (cid:15) (cid:15) D ( m , p ) ∧ D ( n , q ) ⊛ (cid:15) (cid:15) g ∧ g / / D ( m , p ) ∧ D ( n , q ) ⊛ (cid:15) (cid:15) γ ( g ) ∗ ∧ δ ( g ) ∗ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ D ( mn , pq ) g / / D ( mn , pq ) ( γ ⊗ δ )( g ) ∗ ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ D ( mn , pq ) ( α ⊗ β )( g − ) ∗ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ P g / / D ( mn , pq )) The central square commutes since D is a Cat ( G V ∗ )-category of operators, makingthe displayed functors ⊛ internal to G V ∗ and therefore equivariant. The left andright trapezoids commute because ⊛ is strict when composing with morphisms in Π according to condition (2) of Definition 5.7. The upper and lower trapezoidscommute by definition, as in (6.8).The pseudofunctoriality constraint P D ( n β , p γ ) ∧ P D ( r ζ , s η ) ∧ P D ( m α , n β ) ∧ P D ( q δ , r ζ ) ⊛ ∧ ⊛ / / id ∧ t ∧ id ∼ = (cid:15) (cid:15) ✕✕✕✕ (cid:6) (cid:14) ϑ P D ( nr β ⊗ ζ , ps γ ⊗ η ) ∧ P D ( mq α ⊗ δ , nr β ⊗ ζ ) ◦ (cid:15) (cid:15) P D ( n β , p γ ) ∧ P D ( m α , n β ) ∧ P D ( r ζ , s η ) ∧ P D ( q δ , r ζ ) ◦∧◦ (cid:15) (cid:15) P D ( m α , p γ ) ∧ P D ( q δ , s η ) ⊛ / / P D ( mq α ⊗ δ , ps γ ⊗ η ) is just that of D at the corresponding objects with trivial action (see (5.8)). Theequivariance of ϑ with respect to the original action on D and the conditions on ϑ from Remark 5.10 combine to show that ϑ prewhiskered with P g is equal to ϑ postwhiskered with P g , as needed.The symmetry τ is prolonged similarly. The 1-cell at the object ( m α , p β ) is thepermutation τ m,p considered as a 1-cell in P D . The pseudonaturality constraint ˆ τ (see (5.9)) is given by that on D . One needs to check that this V ∗ -transformationis G -equivariant with respect to the prolonged action. This follows from the equiv-ariance of ˆ τ with respect to the original action on D , the compatibility of ˆ τ with ϑ ,condition (2) of Definition 5.7, and the fact that ˆ τ restricted to Π is the identity. (cid:3) The construction is functorial with respect to pseudo-commutative morphismsof pseudo-commutative categories of operators over F . Theorems 5.12 and 6.14have the following corollary. Corollary 6.16. If O is a pseudo-commutative operad, then D G ( O ) is pseudo-commutative category of operators over F G . Algebras and pseudoalgebras over categories of operators over F G . Just as for categories of operators over F , Definitions 4.18, 4.23, and 4.26 special-ize to define a 2-category of algebras, pseudomorphisms, and transformations forcategories of operators D G over F G . Notation 6.17.
Let D G be a Cat ( G V ∗ )- CO over F G . We denote by D G - Alg ps the2-category of strict D G -algebras, ( D G , Π G )-pseudomorphisms, and D G -transformations.This 2-category was denoted by D G - Alg ps Π G in Notation 4.27. Just as in Notation 5.14we fix B = Π G and drop the subscript from the notation.Again, we do not discuss general D G -pseudoalgebras here, leaving such con-sideration for [14]. However, we will need pseudoalgebras in the special case of D G = F G starting in Section 7. Recall from Notation 4.27 that we have the 2-category F G - PsAlg of weak F G -pseudoalgebras, weak F G -pseudomorphisms, and F G -transformations. Remark 6.18.
We comment on the choice of weak pseudoalgebras. This choiceis already essential nonequivariantly. Look back at (1.2), but take G = e . Withmore effort, we could have started with O -pseudoalgebras, as defined in [16]; thefunctor R would then land in D -pseudoalgebras that are strict over Π. However, ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 47 the section ζ : F −→ D (see Proposition 7.4) loses strictness with respect to Π, asexplained in Remark 7.5, so that whether the domain of ζ ∗ is taken to be D - Alg ps or some 2-category of D -pseudoalgebras, its target must still be F - PsAlg . It takesstrict D -algebras only to weak F -pseudoalgebras.6.5. The symmetric multicategory of D G -algebras. Let D G be a pseudo-commutative category of operators over F G . We define the multicategory Mult ( D G )of D G -algebras, which amounts to defining the k -ary morphisms. The definition isexactly like Definition 5.15, hence we refer the reader there for details. Again,we set it up to have its objects be D G -algebras, although with only slightly morework we could equally well have set it up to have its objects be D G -pseudoalgebras.Remember that we write θ : D G ( m α , n β ) ∧ X ( m α ) −→ X ( n β )for the action of D G on a D G -algebra X . For a k -tuple of finite G -sets m α i i , wewrite m α for the finite G -set with m = m · · · m k and α = α ⊗ · · · ⊗ α k .Recall from Definition 4.29 that given D G -algebras X , . . . , X k , we have the ex-ternal smash product X ∧ . . . ∧X k , which is a D ∧ kG -algebra. For a D G -algebra Y ,we have the D ∧ kG -pseudoalgebra Y ◦ ⊛ k , which is strict over Π ∧ kG . Definition 6.19.
We define a symmetric multicategory
Mult ( D G ) of D G -algebras.The objects are the D G -algebras. For D G -algebras X i , 1 ≤ i ≤ k , and Y , a k -arymorphism X −→ Y consists of a ( D ∧ kG , Π ∧ kG )-pseudomorphism F : X ∧ . . . ∧X k / / /o/o/o Y ◦ ⊛ k . Composition and the symmetric action are specified as in Definition 5.15.Unpacking the definition, a k -ary morphism F = ( F, δ ) consists of G V ∗ -functors F : X ( m α ) ∧ · · · ∧ X k ( m α k k ) −→ Y ( m α ) , together with invertible G V ∗ -transformations δ as in the following diagram, in which1 ≤ i ≤ k .(6.20) V i D G ( m α i i , n β i i ) ∧ V i X i ( m α i i ) id ∧ F / / t ∼ = (cid:15) (cid:15) ✎✎✎✎ (cid:3) (cid:11) δ V i D G ( m α i i , n β i i ) ∧ Y ( m α ) ⊛ ∧ id (cid:15) (cid:15) V i D G ( m α i i , n β i i ) ∧ X i ( m α i i ) V i θ (cid:15) (cid:15) D G ( m α , n β ) ∧ Y ( m α ) θ (cid:15) (cid:15) V i X i ( n β i i ) F / / Y ( n β ) . The strictness over Π ∧ kG is encoded by requiring δ to be the identity if all factors D G are restricted to Π G . These transformations must satisfy coherence with respectto composition in D ∧ kG , details of which can be found in Section 11.3.We have the following analogue of Theorem 5.20; its proof is essentially the same. Theorem 6.21.
Let (Ψ , µ ) : D G −→ E G be a map of pseudo-commutative cate-gories of operators over F G . Then pulling back along Ψ induces a (symmetric)multifunctor Ψ ∗ : Mult ( E G ) −→ Mult ( D G ) . The symmetric multifunctor P . Let D be a Cat ( G V ∗ )-category of oper-ators over F and D G = P D be its prolonged category of operators over F G , asdefined in Construction 6.7. Define U : D G - Alg ps −→ D - Alg ps by restricting alongthe inclusion D ⊂ D G . Then U has a left adjoint prolongation functor on the levelof algebras, P : D - Alg ps −→ D G - Alg ps . For the subcategory of strict maps, this is the categorical analogue of the prolon-gation functor from [39, § P to ( D , Π)-pseudomorphisms in Theorem 6.23 below.On objects, P ( X )( n α ) ∈ Cat ( G V ∗ ) is defined by letting P ( X )( m α ) be a copy of X ( m ), but with the action of g ∈ G , denoted P g when α is understood, defined tobe the composite X ( m ) g / / X ( m ) α ( g ) ∗ / / X ( m ) . Here α ( g ) ∗ : X ( m ) −→ X ( m ) is given by the action of Π on X . The enrichedfunctor X takes the morphisms of Π, which are G -fixed, to G -equivariant functors.It follows that we can equivalently write P g as the composite(6.22) X ( m ) α ( g ) ∗ / / X ( m ) g / / X ( m ) . The action G V ∗ –functor P θ : D G ( m α , n β ) ∧ P X ( m α ) −→ P X ( n β )is defined to be θ : D ( m , n ) ∧ X ( m ) −→ X ( n ) . The following diagram shows that P θ is equivariant because θ is equivariant, asdisplayed in the middle square. D ( m , n ) ∧ X ( m ) θ (cid:15) (cid:15) α ( g − ) ∗ ∧ α ( g ) ∗ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ P g ∧ P g / / D ( m , n ) ∧ X ( m ) θ (cid:15) (cid:15) D ( m , n ) ∧ X ( m ) θ (cid:15) (cid:15) g ∧ g / / D ( m , n ) ∧ X ( m ) θ (cid:15) (cid:15) β ( g ) ∗ ∧ id ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ X ( n ) g / / X ( n ) β ( g ) ∗ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ X ( n ) ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ P g / / X ( n ) Since the action θ is compatible with composition in D , the left and right trapezoidscommute, the left one using that α ( g − ) ∗ ◦ α ( g ) ∗ = id.With these preliminaries, we have the following result. ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 49 Theorem 6.23.
The functor P : D - Alg ps −→ D G - Alg ps extends to a (symmetric)multifunctor P : Mult ( D ) −→ Mult ( D G ) . Proof.
Since the values of P X at objects in D G are the values of X but with a new G -action, the idea of the proof is to show that the data of a map of D -algebrasremains G -equivariant with respect to the new action.Thus let ( F, δ ) : ( X , . . . , X k ) −→ Y be a k -ary morphism of D -algebras. Thismeans that we are given G V ∗ -functors F : X ( m ) ∧ · · · ∧ X k ( m k ) −→ Y ( m )and invertible G V ∗ -transformations δ is as in (5.17). We define P ( F, δ ) = ( P F, P δ )as follows. For m α , . . . , m α k k , with m = m · · · m k and α = α ⊗ · · · ⊗ α k , the map P F : P X ( m α ) ∧ · · · ∧ P X k ( m α k k ) −→ P Y ( m α ) , is given by F . Similarly, we define P δ in the diagram (6.20) to be δ in the underlyingdiagram (5.17).We must check that these give G V ∗ -functors and G V ∗ -transformations, respec-tively, that is, that they are equivariant with respect to the prolonged action. When k = 1, this gives the promised definition of P on pseudomorphisms of D -algebras.We first check that P F is equivariant. For ease of notation, we consider the case k = 2. Recall the G -action on P X ( m α ) given in (6.22). Then to check that P X ( m α ) ∧ P X ( m α ) P F −−→ P Y ( m α )is G -equivariant, it suffices to show that the diagram X ( m ) ∧ X ( m ) F / / α ( g ) ∗ ∧ α ( g ) ∗ (cid:15) (cid:15) Y ( m ) α ( g ) ∗ (cid:15) (cid:15) X ( m ) ∧ X ( m ) F / / g ∧ g (cid:15) (cid:15) Y ( m ) g (cid:15) (cid:15) X ( m ) ∧ X ( m ) F / / Y ( m )commutes. The top square commutes since F restricts to a strict transformationof Π-functors (see Definition 4.23). The bottom square commutes since F is equi-variant with respect to the original action.It remains to check that P δ is equivariant with respect to the prolonged action.By Remark 6.1, this is done by proving that prewhiskering δ with P g is equal topostwhiskering it. We illustrate by taking k = 1. Thus we return to Definition 4.23.Let X and Y be D -algebras and let F : X / / /o/o/o Y be a ( D , Π)-pseudomorphism.
The 2-cell D ( m , n ) ∧ X ( m ) ∧ F / / β ( g ) ∗ α ( g − ) ∗ ∧ α ( g ) ∗ (cid:15) (cid:15) D ( m , n ) ∧ Y ( m ) β ( g ) ∗ α ( g − ) ∗ ∧ α ( g ) ∗ (cid:15) (cid:15) D ( m , n ) ∧ X ( m ) ∧ F / / g ∧ g (cid:15) (cid:15) D ( m , n ) ∧ Y ( m ) g ∧ g (cid:15) (cid:15) D ( m , n ) ∧ X ( m ) ∧ F / / θ (cid:15) (cid:15) ✓✓✓✓ (cid:5) (cid:13) δ D ( m , n ) ∧ Y ( m ) θ (cid:15) (cid:15) X ( n ) F / / Y ( n ) is D ( m , n ) ∧ X ( m ) ∧ F / / β ( g ) ∗ α ( g − ) ∗ ∧ α ( g ) ∗ (cid:15) (cid:15) D ( m , n ) ∧ Y ( m ) β ( g ) ∗ α ( g − ) ∗ ∧ α ( g ) ∗ (cid:15) (cid:15) D ( m , n ) ∧ X ( m ) ∧ F / / θ (cid:15) (cid:15) ✓✓✓✓ (cid:5) (cid:13) δ D ( m , n ) ∧ Y ( m ) θ (cid:15) (cid:15) X ( n ) F / / g (cid:15) (cid:15) Y ( n ) g (cid:15) (cid:15) X ( n ) F / / Y ( n ) , by the equivariance of δ with respect to the original action, and this also agreeswith the 2-cell D ( m , n ) ∧ X ( m ) ∧ F / / θ (cid:15) (cid:15) ✓✓✓✓ (cid:5) (cid:13) δ D ( m , n ) ∧ Y ( m ) θ (cid:15) (cid:15) X ( n ) F / / β ( g ) ∗ (cid:15) (cid:15) Y ( n ) β ( g ) ∗ (cid:15) (cid:15) X ( n ) F / / g (cid:15) (cid:15) Y ( n ) g (cid:15) (cid:15) X ( n ) F / / Y ( n ) by the compatibility of δ with composition and the strictness of δ with respect toΠ. The equivariance of P δ for k > δ in asimilar way. The diagrams are larger, but the verification is essentially the same.Since P F and P δ are just F and δ on the underlying V ∗ -categories, it followsthat P respects composition, the identity, and the Σ-action. (cid:3) Definition 6.24.
We define R G to be the composite O - Alg ps R G & & ▼▼▼▼▼▼▼▼▼▼ R / / D - Alg ps P (cid:15) (cid:15) D G - Alg ps Corollary 6.25.
The functor R G : O - Alg ps −→ D G - Alg ps extends to a symmetricmultifunctor. From D G -algebras to F G -pseudoalgebras We recall the notion of an E ∞ G -operad in G U and in Cat ( G U ) in Section 7.1.For a category of operators D G arising from a chaotic E ∞ G -operad O , we pro-duce a pseudofunctor ζ G : F G / / /o/o/o D G that is a section to ξ G in Proposition 7.4.Finally, in Theorem 7.13, we show that pulling back along ζ G defines a symmetricmultifunctor from D G -algebras to F G -pseudoalgebras. ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 51 Throughout this section, we restrict attention to the topological case V = U andthus G V = G U .7.1. E ∞ G -operads. So far, our chaotic operads have been quite general. We nowrestrict attention to a chaotic E ∞ G -operad O of G U -categories and its associatedcategory of operators D G = D G ( O ) over F G . Definition 7.1.
An operad O in G U is an E ∞ G -operad if for all n ≥ G × Σ n , the fixed point space O ( n ) Λ is contractible if Λ ∩ Σ n = { e } and is empty otherwise.An operad O in Cat ( G U ) is an E ∞ G -operad if the operad B O obtained byapplying the classifying space functor levelwise is an E ∞ G -operad of G -spaces.The condition on fixed-points implies that for an E ∞ G -operad in G U , the space O ( n ) is a universal principal ( G, Σ n )-bundle. Algebras over E ∞ G -operads, are, upto group completion, equivariant infinite loop spaces with deloopings with respectto all finite-dimensional G -representations, and thus give rise to genuine G -spectra.For more background and examples we refer the reader to [13, Section 2.1].The following result shows how chaotic categories are useful in this context. Proposition 7.2.
Let O be a chaotic operad in Cat ( G U ) . Then O is an E ∞ G -operad if and only if for all n ≥ and all subgroups Λ of G × Σ n , the fixed pointobject space ( Ob O ( n )) Λ is non-empty if Λ ∩ Σ n = { e } and is empty otherwise.Proof. As noted in [16, Remark 1.15], the classifying space of a non-empty chaotic U -category is contractible. Thus, the statement follows by noting that if O ischaotic, then O ( n ) Λ is also chaotic. (cid:3) The section map ζ G from F G to D G . Recall that D G comes equipped withfunctors ι G : Π G −→ D G and ξ G : D G −→ F G such that ξ G ◦ ι G is the inclusion. We here define an (equivariant) section ζ G : F G / / /o/o/o D G to ξ G . Definition 7.3. A pseudomorphism ν : D G / / /o/o/o E G of CO s over F G is a G U ∗ -pseudofunctor over F G and under Π G . Proposition 7.4.
Let O be a chaotic E ∞ G -operad in Cat ( G U ) and let D G = D G ( O ) . Then there exists a pseudomorphism ζ G : F G / / /o/o/o D G of CO s over F G and an invertible G U ∗ -pseudotransformation D G ξ G ! ! ❉❉❉❉❉❉❉❉ id / / ✤✤ ✤✤ (cid:11) (cid:19) χ D G . F G ζ G < < <|<|<|<|<| Proof.
On objects, we have no choice: ζ G is the identity. More generally, on Π G we must take ζ G = ι G . Given finite G -sets m α and n β , we must specify a based G -equivariant function ζ G : F G ( m α , n β ) −→ Ob D G ( m α , n β ) . To define an equivariant function, it suffices to specify the function on each G -orbit.Moreover, an equivariant function out of an orbit is completely determined by itsvalue at any point in the orbit. We thus choose, for each m α and n β , a point ofeach G -orbit of the G -set F G ( m α , n β ).Let f ∈ F G ( m α , n β ) be such a chosen element. Let H ≤ G be the stabilizer of f . The section ζ G must send f to an H -fixed object of the G -category D G ( m α , n β )that is in the component Q ≤ j ≤ k O ( f − ( j )) of f . Since O is an E ∞ G -operad, the H -fixed point subset of this component has contractible classifying space, by [39,Theorem 5.4], hence is nonempty. Thus we can choose an H -fixed object ζ G ( f ) inthe component of f . The only exception to such use of choices is that we alreadyknow the definition of ζ G on Π G , so these choices only apply to morphisms of F G that are not in Π G .The claim is that these choices specify an equivariant pseudofunctor ζ G . Theequivariance has been forced by the definition of ζ G : if f is one of our distinguishedpoints, then ζ G ( g · f ) = g · ζ G ( f ). To see the pseudofunctor structure, we mustspecify a G -equivariant natural isomorphism F G ( n β , p γ ) ∧ F G ( m α , n β ) ◦ (cid:15) (cid:15) ζ G ∧ ζ G / / ✖✖✖✖ (cid:7) (cid:15) ϕ D G ( n β , p γ ) ∧ D G ( m α , n β ) ◦ (cid:15) (cid:15) F G ( m α , p γ ) ζ G / / D G ( m α , p γ ) . The component of ϕ at ( h, f ) must be a morphism in D G ( m α , p γ ) of the form ϕ h,f : ζ G ( h ) ◦ ζ G ( f ) −→ ζ G ( h ◦ f ) . Since both of these points in D G ( m α , p γ ) live over h ◦ f ∈ F G ( m α , p γ ) and O ischaotic, there is a unique morphism with the required source and target.We claim that ϕ is G -equivariant. This means that ϕ g · h,g · f = g · ϕ h,f for g ∈ G .This again holds because O is chaotic: there is a unique morphism between anytwo objects, so these two morphisms are necessarily the same. The compatibilityof ϕ with triple composition follows again from the uniqueness of these morphisms.For an object m α , the 1-cell component χ m α : m α −→ ζ G ◦ ξ G ( m α ) = m α is theidentity map. We need to construct the pseudonaturality constraint, which is aninvertible G U ∗ -transformation D G ( m α , n β ) id (cid:15) (cid:15) ζ G ◦ ξ G / / ✏✏✏✏ (cid:4) (cid:12) χ D G ( m α , n β ) ( χ m α ) ∗ (cid:15) (cid:15) D G ( m α , n β ) ( χ n β ) ∗ / / D G ( m α , n β ) . ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 53 For a 1-cell d ∈ D G ( m α , n β ), the object ζ G ◦ ξ G ( d ) is in the same componentof the same G -fixed point subset as d , hence there is a unique morphism χ d : ζ G ◦ ξ G ( d ) −→ d , and this assignment is continuous and equivariant. The uniquenessimplies naturality and the required compatibility with ϕ . That and the evidentinverse isomorphism lead to the conclusion. (cid:3) Remark 7.5.
Although the section ζ G is strictly functorial when restricted to Π G ,it is not when only one of the morphisms of a composite in F G is in Π G , even when G = e . Let p n : n −→ be the based function that sends j to 1 for 1 ≤ j ≤ n . Then ζ ( p n ) = ( p n ; d ) for some d ∈ O ( n ). For a permutation σ ∈ Σ n , we have p n ◦ σ = p n ,while ζ ( p n ) ◦ ζ ( σ ) = ( p n ; d · σ ). In the cases of interest Σ n acts freely on O ( n ) andwe cannot have ζ ( p n ) ◦ ζ ( σ ) = ζ ( p n ). It is this fact that led us to the distinctionshighlighted in Remark 6.18.Recall the definition of a map between pseudo-commutative categories of opera-tors over F G from Definitions 5.11 and 6.13. We adapt that definition to pseudo-morphisms of CO s over F G . Definition 7.6. A pseudomorphism of pseudo-commutative CO s over F G consistsof a pseudomorphism ν : D / / /o/o/o E of CO s over F G and a G U ∗ -pseudotransformation D ∧ D ν ∧ ν / / /o/o/o ⊛ D (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O ✡✡✡✡ (cid:1) (cid:9) µ E ∧ E ⊛ E (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O D ν / / /o/o/o/o/o E satisfying analogues of all the axioms in Definition 5.11. Proposition 7.7.
Let O be a chaotic G -operad and let D G = D G ( O ) . Then thesection ζ G : F G / / /o/o/o D G is a pseudomorphism of pseudo-commutative CO s.Proof. We must produce a G U ∗ -pseudotransformation µ as above. Since ζ G is theidentity on objects and µ must restrict to the identity on Π G , we take the 1-cellcomponent of µ to be the identity. Given 1-cells f and h in F G , we must producean invertible two-cell µ f,h : ζ G ( f ∧ h ) ∼ = ζ G ( f ) ⊛ ζ G ( h ) , that is the component of a G U ∗ -transformation. Since the specified source andtarget live over f ∧ h in F G , the fact that O is chaotic implies that there is a uniquechoice for µ f,h ; that µ is natural and equivariant, and satisfies all the axioms followsfor the same reason. (cid:3) The symmetric multifunctor ζ ∗ G . We have elected to work with (strict) D G -algebras but, due to Remark 7.5, when we precompose with our section ζ G , weonly produce weak F G -pseudoalgebras; recall from Notation 4.27 that we definedthe objects and morphisms of F G - PsAlg as enriched pseudofunctors from F G to Cat ( G V ∗ ) and pseudotransformations, with no strictness conditions over Π G . Thefollowing proposition follows immediately from the definitions. Proposition 7.8.
Pullback along the section ζ G defines a 2-functor ζ ∗ G : D G - Alg ps −→ F G - PsAlg . Remark 7.9.
Since ζ G restricts to the inclusion ι G on Π G , for any D G -algebra X ,the underlying Π G -algebra of ζ ∗ G X is the underlying Π G -algebra of X . Corollary 7.10.
Let Y be a D G -algebra. Then the pseudonatural isomorphism χ of Proposition 7.4 induces an invertible pseudomorphism χ : Y ∼ = ξ ∗ G ζ ∗ G Y of weak D G -pseudoalgebras (see Definition 4.23) whose components are identitymaps. We define multimorphisms of weak F G -pseudoalgebras following Definition 6.19,but deleting its strictness conditions with respect to Π G . Note that ⊛ is just ∧ on F G . Recall that we are working in Cat ( G U ∗ ) in this section. Definition 7.11.
We define a (symmetric) multicategory
Mult ( F G - PsAlg ) of(weak) F G -pseudoalgebras whose underlying category is F G - PsAlg , as follows.For weak F G -pseudoalgebras X i , 1 ≤ i ≤ k , and Y , a k -ary morphism X −→ Y consists of a F ∧ kG -pseudomorphism F : X ∧ . . . ∧X k / / /o/o/o Y ◦ ∧ k . Composition and the symmetric group action are given by the corresponding past-ing diagrams, as done explicitly in Definition 5.15.The unpacking of this definition is similar to the unpackings given for Definitions5.15 and 6.19, with the caveat that the coherence diagrams in Section 11.3 mustaccount for the pseudofunctoriality constraints of the weak F G -pseudoalgebras. Remark 7.12.
We could define an analogous multicategory F - PsAlg of weakstructures, but we would not have a prolongation multifunctor P : F - PsAlg −→ F G - PsAlg since we would no longer have the compatibility with Π that we used inthe proof of Theorem 6.23. However, naturally occurring examples of F -pseudoalgebrasthat do not arise from use of the section often do have such compatibility with Πand thus can be prolonged to F G -pseudoalgebras.We have the following theorem, whose proof is essentially the same as that ofTheorem 5.20. Recall Notation 4.28. Theorem 7.13.
Pullback along ζ G induces a symmetric multifunctor ζ ∗ G : Mult ( D G ) −→ Mult ( F G - PsAlg ) . Strictification of pseudoalgebras
For clarity about what is general and what is special and also for simplicityof notation, we revert to a general V satisfying our standard assumptions in thissection. The reader may prefer to focus on V = U or V = G U , but equivarianceand topology play no role in this section. ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 55 Power-Lack strictification.
We here specialize a general result of Power andLack [20, 42] (see also [16, Theorem 0.1]) about strictification of pseudoalgebrasover a 2-monad. Let C be a V ∗ -2-category, and recall the 2-categories C - Alg and C - PsAlg (Notation 4.27) of C -algebras and weak C -pseudoalgebras. We prove theexistence of a strictification 2-functor St : C - PsAlg −→ C - Alg . While the generalresult specializes to give everything we need, we prefer to give an independentaccount that translates out from the theory of 2-monads and includes an explicitconstruction. We are implicitly using the existence of an enhanced factorizationsystem on
Cat ( V ∗ ) [16, §
4] to apply Power and Lack’s result to our context.
Theorem 8.1.
Let C be a V ∗ -2-category. The inclusion of -categories J : C - Alg −→ C - PsAlg has a left 2-adjoint St C : C - PsAlg −→ C - Alg . The component of the unit i of the -adjunction is a C -pseudomorphism which is alevelwise equivalence (Definition 4.25). For brevity, we will omit the subscript C and simply write St for the strictificationfunctor when the category C is clear from the context. Proof.
For a weak C -pseudoalgebra ( X , θ, ϕ ), we define the (strict) C -algebra St X as follows.For c in C , the V ∗ -category St X ( c ) has V ∗ -object of objects(8.2) Ob (St X ( c )) = _ b Ob C ( b, c ) ∧ Ob ( X ( b )) , where b ranges over all objects of C .The action of C on X defines a morphism(8.3) θ : Ob (St X ( c )) −→ Ob ( X ( c )) , and this allows us to define the V ∗ -object of morphisms of St X ( c ) as the pullbackdisplayed in the diagram(8.4) Mor (St X ( c )) / / ( T,S ) (cid:15) (cid:15) Mor ( X ( c )) ( T,S ) (cid:15) (cid:15) Ob (St X ( c )) × Ob (St X ( c )) θ × θ / / Ob ( X ( c )) × Ob ( X ( c )) . Composition is induced by composition in X ( c ).To describe the pullback more explicitly, writing elementwise for the sake ofexposition, let( f, x ) ∈ Ob C ( b, c ) ∧ Ob ( X ( b )) and ( f ′ , x ′ ) ∈ Ob C ( b ′ , c ) ∧ Ob ( X ( b ′ )) . Then
Mor (St X ( c )) (cid:16) ( f, x ) , ( f ′ , x ′ ) (cid:17) = Mor ( X ( c )) (cid:16) θ ( f, x ) , θ ( f ′ , x ′ ) (cid:17) . The action map St θ : C ( b, c ) ∧ St X ( b ) −→ St X ( c ) descends to the smash product from the map on the product given on objects by(St θ )( h, ( f, x )) = ( h ◦ f, x ) for h ∈ C ( b, c ).For a morphism α : ( f, x ) −→ ( f ′ , x ′ ) in St X ( b ) and λ : h −→ h ′ in C ( b, c ), thecorresponding morphism St θ ( h, ( f, x )) −→ St θ ( h ′ , ( f ′ , x ′ )) is defined to be θ ( h ◦ f, x ) ϕ − −−→ θ ( h, θ ( f, x )) θ ( λ,α ) −−−−→ θ ( h ′ , θ ( f ′ , x ′ )) ϕ −→ θ ( h ′ ◦ f, x ) . Then St θ gives St X a strict C -algebra structure by the strict functoriality of com-position in C .For a C -pseudomorphism ( F, δ ) : X / / /o/o/o Y , we define St( F, δ ) : St
X −→ St Y byletting St( F, δ ) c : St X ( c ) −→ St Y ( c ) be the functor sending ( f, x ) to ( f, F x ) and α : ( f, x ) −→ ( g, y ) to the composite θ Y ( f, F x ) δ − −−→ F ( θ X ( f, x )) F α −−→ F ( θ X ( g, y )) δ −→ θ Y ( g, F y ) . It is straightforward to check that these form the components of a strict C -morphism.We can similarly define the action of St on C -transformations.For a weak C -pseudoalgebra ( Y , θ, ϕ ), we define C -pseudomorphisms i : Y / / /o/o/o St Y and m : St Y / / /o/o/o Y as follows. The component V ∗ -functor i c : Y ( c ) −→ St Y ( c ) is given by i c ( y ) =(id c , y ) on objects y of Y ( c ) and i c ( γ ) = γ on morphisms γ of Y ( c ). The latter makessense since θ (id c , y ) = y . The component of the pseudonaturality V ∗ -transformation C ( b, c ) ∧ Y ( b ) id ∧ i b / / θ (cid:15) (cid:15) ✑✑✑✑ (cid:4) (cid:12) i b,c C ( b, c ) ∧ St Y ( b ) St θ (cid:15) (cid:15) Y ( c ) i c / / St Y ( c )at ( f, y ) is the morphism ( f, y ) −→ (id c , θ ( f, y )) in St Y ( c ) corresponding to theidentity map of θ ( f, y ) in Y ( c ).The component V ∗ -functors m c : St Y ( c ) −→ Y ( c ) are given by (8.3) on the V ∗ -object of objects and by the top horizontal arrow in (8.4) on the V ∗ -object ofmorphisms. The pseudonaturality constraint C ( b, c ) ∧ St Y ( b ) id ∧ m b / / St θ (cid:15) (cid:15) ✑✑✑✑ (cid:4) (cid:12) ˜ ϕ C ( b, c ) ∧ Y ( b ) θ (cid:15) (cid:15) St Y ( c ) m c / / Y ( c )is induced by the invertible V ∗ -transformation ϕ that witnesses the C -pseudoalgebrastructure of Y .As is easily checked directly, i and m are inverse equivalences in C - PsAlg , soin particular, i is a level equivalence. The map i is 2-natural with respect to C -pseudomorphisms and C -transformations and is the unit of the adjunction. If Y isa strict C -algebra and thus of the form Y = J X , then m is a strict C -morphism, and ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 57 moreover, it is 2-natural with respect to strict C -morphisms and C -transformations;it is the counit of the adjunction. (cid:3) Remark 8.5.
The C -pseudomorphism m : St Y / / /o/o/o Y of the proof of Theorem 8.1is not strictly natural with respect to C -pseudomorphisms of weak C -pseudoalgebras.With slightly extra work, one can prove that it gives the components of a pseudo-natural transformation m : J St = ⇒ id of endo-2-functors of C - PsAlg . As it is notnecessary for our work, we shall not pursue this route. However, we will use the fact,noted in the proof, that m c : St Y ( c ) −→ Y ( c ) is an equivalence of V ∗ -categories.Recall Notation 4.28. We shall apply the following result to an iterated monoidalproduct C ∧ k −→ C in Section 8.2 and to ξ G : D G −→ F G in Section 10.2. Lemma 8.6.
Let D and C be V ∗ -2-categories and let ξ : D −→ C be a V ∗ -2-functor.Given a C -pseudoalgebra Y , there is a D -morphism ψ = ψ ξ : St D ( ξ ∗ Y ) −→ ξ ∗ (St C Y ) that is 2-natural with respect to C -pseudomorphisms and C -transformations. More-over, the diagram ξ ∗ Y i D z z z: z: z: z: z: ξ ∗ i C $ $ $d$d$d$d$d St D ( ξ ∗ Y ) ψ / / ξ ∗ (St C Y ) commutes, hence ψ is a levelwise equivalence.Proof. We can specify ψ as the D -morphism corresponding to the D -pseudomorphism ξ ∗ i C : ξ ∗ Y / / /o/o/o ξ ∗ (St C Y )under the adjunction of Theorem 8.1. The claim about 2-naturality then followsfrom the 2-naturality of i . The commutativity of the diagram can be verified directlyfrom the definition. (cid:3) Remark 8.7.
For later use, we give an explicit description of ψ in terms of elements.For an object d ∈ D , ψ d : St D ( ξ ∗ Y )( d ) −→ ξ ∗ (St C Y )( d ) sends an object ( f, y ), with f : d ′ → d in D and y ∈ Y ( ξ ( d ′ )), to the object ( ξ ( f ) , y ).The following lemma records the compatibility of the morphism ψ of Lemma 8.6with composition of V ∗ -2-functors; it follows directly from the definitions. Lemma 8.8.
Let ν : E −→ D and ξ : D −→ C be V ∗ -2-functors. Then the diagram St E (( ξν ) ∗ Y ) ψ ( ξν ) / / ( ξν ) ∗ (St C Y )St E ( ν ∗ ξ ∗ Y ) ψ ν / / ν ∗ (St D ( ξ ∗ Y )) ν ∗ ψ ξ / / ν ∗ ξ ∗ (St C Y ) commutes for every C -pseudoalgebra Y . The following observation about the interaction of strictification with the exter-nal smash product (Definition 4.29) generalizes [32, Lemma 3.5].
Lemma 8.9.
Let C and D be V ∗ -2-categories, X a C -pseudoalgebra and Y a D -pseudoalgebra. Then there is a canonical isomorphism St C ∧ D ( X ∧ Y ) ∼ = (St C X ) ∧ (St D Y ) , which is -natural with respect to the respective pseudomorphisms and pseudotransformations.In particular, up to composing with this canonical isomorphism, for a C -pseudomorphism E : X / / /o/o/o X ′ and an D -pseudomorphism F : Y / / /o/o/o Y ′ , the C ∧ D -morphism St C ∧ D ( E ∧ F ) corresponds to (St C E ) ∧ (St D F ) .Proof. On the level of objects, the identification follows from (2.8) and (8.2). Againwriting elementwise, it is just the twist that sends an object (( e, f ) , ( x, y )) ofSt C ∧ D ( X ∧ Y )( c, d ) to the object (( e, x ) , ( f, y )) of (St C X )( c ) ∧ (St D Y )( d ).Note that since the V ∗ -functor m C ∧ m D : (St C X ( c )) ∧ (St D Y ( d )) → X ( c ) ∧ Y ( d )is an equivalence (Remark 8.5), it is fully faithful [44, Lemma 4.17], in the sensethat the diagram Mor (cid:16) (St C X ( c )) ∧ (St D Y ( d )) (cid:17) / / ( T,S ) (cid:15) (cid:15) Mor ( X ( c ) ∧ Y ( d )) ( T,S ) (cid:15) (cid:15) Ob (cid:16) (St C X ( c )) ∧ (St D Y ( d )) (cid:17) × / / Ob ( X ( c ) ∧ Y ( d )) × is a pullback square which is isomorphic to the pullback square in (8.4) that defines Mor (St C ∧ D ( X ∧ Y )). This shows that the V ∗ -categories St C ∧ D ( X ∧ Y )( c, d ) and(St C X )( c ) ∧ (St D Y )( d ) are isomorphic. One can check that these isomorphismsrespect the action of C ∧ D , thus proving the result. (cid:3) We record the relationship between the 2-natural transformation ψ of Lemma 8.6and the canonical isomorphism of Lemma 8.9. Lemma 8.10.
Let C , C ′ , D , and D ′ , be V ∗ -2-categories, let ξ : C ′ −→ C and ζ : D ′ −→ D be V ∗ -2-functors, and let X be a C -pseudoalgebra and Y be a D -pseudoalgebra. Then the following diagram commutes, where the unnamed isomor-phisms are those of Lemma 8.9. St C ′ ∧ D ′ (cid:16) ( ξ ∧ ζ ) ∗ ( X ∧ Y ) (cid:17) ψ ξ ∧ ζ / / ( ξ ∧ ζ ) ∗ St C ∧ D ( X ∧ Y ) ∼ = (cid:15) (cid:15) St C ′ ∧ D ′ (cid:16) ( ξ ∗ X ) ∧ ( ζ ∗ Y ) (cid:17) ∼ = (cid:15) (cid:15) ( ξ ∧ ζ ) ∗ (St C X ∧ St D Y )(St C ′ ξ ∗ X ) ∧ (St D ′ ζ ∗ Y ) ψ ξ ∧ ψ ζ / / ( ξ ∗ St C X ) ∧ ( ζ ∗ St D Y ) ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 59 The extension of strictification to a multifunctor.
We now assume that( C , I, ⊛ , τ ) is a permutative V ∗ -2-category (Definition 4.30). We extend St to amultifunctor from the multicategory Mult ( C - PsAlg ) of (weak) C -pseudoalgebrasto the multicategory Mult ( C - Alg ) of strict C -algebras and strict multilinear maps.The former is defined by replacing F G by C in Definition 7.11, and we now definethe latter. Definition 8.11.
The multicategory
Mult ( C - Alg ) is constructed by using thesymmetric monoidal structure on C - Alg given by Day convolution along iterationsof the monoidal product ⊛ on C . Reinterpreting this externally, via the universalproperty of Day convolution, we see that for strict C -algebras X , . . . , X k and Y , a k -ary morphism is given by a strict C ∧ k -morphism F : X ∧ · · · ∧ X k −→ Y ◦ ⊛ k . We remark that the definition above differs from Definitions 5.15 and 6.19 inthat the multimorphisms are pseudotransformations in those cases, and strict here.
Theorem 8.12.
The strictification functor St induces a (non-symmetric) multi-functor St :
Mult ( C - PsAlg ) −→ Mult ( C - Alg ) . Proof.
For a C -pseudoalgebra X , we set St X = St C X . For multimorphisms, weuse the 2-functor St C ∧ k , which for brevity we denote by St k , to strictify C ∧ k -pseudomorphisms. More precisely, recall that a k -ary morphism in Mult ( C - PsAlg )is given by a C ∧ k -pseudomorphism F : X ∧ · · · ∧ X k / / /o/o/o Y ◦ ⊛ k . We define the k -ary morphism St( F ) in Mult ( C ) as the compositeSt X ∧ · · · ∧ St X k ∼ = / / St k ( X ∧ · · · ∧ X k ) St k ( F ) / / St k ( Y ◦ ⊛ k ) ψ / / (St Y ) ◦ ⊛ k , where the unnamed isomorphism is that of Lemma 8.9, and the map ψ is the onedefined in Lemma 8.6.Note that on 1-ary morphisms, St is just the 2-functor St C , so in particular thisassignment sends the identity to itself.It remains to prove that St preserves multicomposition. Let F : Y ∧ · · · ∧ Y k / / /o/o/o Z ◦ ⊛ k be a C ∧ k -pseudomorphism and, for 1 ≤ r ≤ k , let E r : X r, ∧ · · · ∧ X r,j r / / /o/o/o Y r ◦ ⊛ j r be a C ∧ j r -pseudomorphism. The composite in Mult ( C - PsAlg ) is given by thepasting diagram in (5.16). In terms of the external smash product of Definition 4.29and the pullback of Notation 4.28, this C ∧ j -pseudomorphism can be expressed asthe composite of C j -pseudomorphisms V r,i X r,i V r E r / / /o/o/o/o/o V r ( ⊛ j r ) ∗ Y r = ( V r ⊛ j r ) ∗ ( V r Y r ) ( V r ⊛ jr ) ∗ F / / /o/o/o/o/o ( V r ⊛ j r ) ∗ ( ⊛ k ) ∗ Z = ( ⊛ j ) ∗ Z . Consider the following diagram of C ∧ j -morphisms. V r,i St X r,i ∼ = / / ∼ = (cid:15) (cid:15) V r St j r (cid:16)V i X r,i (cid:17) V r St jr ( E r ) / / ∼ = v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠ V r St j r (cid:16) ( ⊛ j r ) ∗ Y r (cid:17) V r ψ ⊛ jr (cid:15) (cid:15) ∼ = v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ St j (cid:16)V r,i X r,i (cid:17) St j (cid:16) V r E r (cid:17) (cid:15) (cid:15) V r ( ⊛ j r ) ∗ St Y r St j (cid:16)V r ( ⊛ j r ) ∗ Y r (cid:17) ( V r ⊛ j r ) ∗ (cid:16)V r St Y r (cid:17) ∼ = (cid:15) (cid:15) St j (cid:16) ( V r ⊛ j r ) ∗ ( V r Y r ) (cid:17) St j (cid:0) ( V r ⊛ jr ) ∗ F (cid:1) (cid:15) (cid:15) ψ ( V ⊛ jr ) / / ( V r ⊛ j r ) ∗ St k (cid:16)V r Y r (cid:17) ( V r ⊛ jr ) ∗ St k F (cid:15) (cid:15) St j (cid:16) ( V r ⊛ j r ) ∗ ( ⊛ k ) ∗ Z (cid:17) ψ ( V ⊛ jr ) / / ( V r ⊛ j r ) ∗ St k (cid:16) ( ⊛ k ) ∗ Z (cid:17) ( V r ⊛ jr ) ∗ ψ ⊛ k (cid:15) (cid:15) St j (cid:16) ( ⊛ j ) ∗ Z (cid:17) ψ ⊛ j / / ( ⊛ j ) ∗ St Z ( V r ⊛ j r ) ∗ ( ⊛ k ) ∗ St Z Strictifying E , . . . , E r and F and then composing is equal to going around clock-wise. Using that St j is a 2-functor, we get that going around counter-clockwise isequal to strictifying the composite. The diagram commutes; indeed, going fromtop to bottom, the regions commute by associativity of ∧ in Cat ( V ∗ ), Lemma 8.9,Lemma 8.10, naturality of ψ , and Lemma 8.8, respectively. (cid:3) Remark 8.13.
This proof depends crucially on the fact that the monoidal producton C is a strict V ∗ -2-functor and not just a pseudofunctor. It does not work forour pseudo-commutative categories of operators D or D G . This is the crux of whythe route in this paper is less categorically intensive than the monadic route of [28],which simultaneously strictifies and transfers structure from D G to F G .8.3. St is not a symmetric multifunctor. As stated in Theorem 8.12, St isnot a symmetric multifunctor. We explain why in this parenthetical subsection.We consider the case when k = 2, with σ the non-trivial element of Σ . Sincethe problem already appears nonequivariantly, we take G = e and specialize to C = F . Thus let F : X ∧ X / / /o/o/o Y be a (weak) F -pseudomorphism between(weak) F -pseudoalgebras. Following (5.18), for an object ( m , p ) of F ∧ F , the1-cell component of F σ is the composite X ( m ) ∧ X ( p ) t / / X ( p ) ∧ X ( m ) F / / Y ( pm ) Y ( τ p,m ) / / Y ( mp ) . ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 61 We claim that the 1-cell components of St(
F σ ) and (St F ) σ do not agree. Foran object ( m , p ) of F ∧ F , these are V ∗ -functorsSt X ( m ) ∧ St X ( p ) −→ St Y ( mp ) . We compare them at the level of objects, writing elementwise. An object of thesource has the form (cid:16) ( f , x ) , ( f , x ) (cid:17) where f : q −→ p and f : n −→ m are morphisms of F and x and x are objectsof X ( q ) and X ( n ), respectively.Then St( F ) σ sends (cid:16) ( f , x ) , ( f , x ) (cid:17) to (cid:16) ( τ p,m ◦ ( f ∧ f ) , F ( x , x ) (cid:17) . We canrewrite the output as (cid:16) ( f ∧ f ) ◦ τ q,n , F ( x , x ) (cid:17) . On the other hand, St(
F σ ) sends (cid:16) ( f , x ) , ( f , x ) (cid:17) to (cid:16) ( f ∧ f ) , θ ( τ q,n , F ( x , x )) (cid:17) . We conclude that the multifunctor St is not symmetric.
Remark 8.14.
This failure of symmetry is forced by our need to use weak pseudo-structure in the target of ζ ∗ G . If we instead use pseudofunctors which are strictrelative to Π when we strictify, for example using the generalized strictificationtheorem given in [14], then we do have symmetry. Symmetry is also studied in the2-monadic context in [28], where the problem is entirely different.9. From F G -algebras in Cat to G -spectra In this section, we describe how we pass from categorical to topological F G -algebras and D G -algebras, and then to G -spectra, keeping track of multiplicativestructure. Nonequivariantly, F -spaces (aka Γ-spaces) were introduced by Segal inhis treatment of infinite loop space theory. These generalize to F G - G -spaces, whichare the input of the equivariant version of the Segal infinite loop space machine. Fora detailed treatment of F G - G -spaces, we refer the reader to [39], and for a treatmentof its symmetric monoidal structure to [15]. Topological categories of operators D and D -spaces were introduced in [37] as an intermediary between F -spaces andoperadic algebras in the proof of the uniqueness of infinite loop space machines.The topological equivariant analogues, D G -spaces, are treated extensively in [39] inthe comparison of equivariant infinite loop space machines.In Section 9.1, we discuss the classifying space functor multiplicatively. We recallthe equivariant Segal machine in Section 9.2. Using that the classifying space func-tor and the Segal machine are both lax monoidal, we restate and prove Theorem Aas Theorem 9.13. In effect, it gives a multiplicative equivariant infinite loop spacemachine starting from operadic categorical input.Some technicalities ensuring that our passage from categorical to space-levelinput is homotopically well-behaved are postponed to Section 9.3. The point is just to give conditions on the categorical input that ensure that the output F G - G -spaces have nondegenerate basepoints. We briefly discuss a related open questionabout Day convolution in Section 9.4. The brief Section 9.5 shows how to obtainhomotopies between maps of G -spectra from operadic categorical input.9.1. The multifunctor B . In order to construct equivariant spectra from O -algebras in a multiplicative way, we need to understand the multiplicative propertiesof the classifying space functor B .The classifying space functor B does not commute with smash products in gen-eral. However, we have the following result, which allows us to use B to changeenrichment. Recall Definition 2.6. Proposition 9.1.
The classifying space functor B : Cat ( G U ∗ ) −→ G U ∗ is laxsymmetric monoidal.Proof. The map B C × B D ∼ = B ( C × D ) −→ B ( C ∧ D )sends the subspace B C ∨ B D to the basepoint and therefore induces a based map B C ∧ B D −→ B ( C ∧ D ) . (cid:3) Definition 9.2.
Let D G be a Cat ( G U ∗ )-category of operators over F G , as definedin Definition 6.5. Let D topG denote the category enriched in G U ∗ obtained by apply-ing B to morphism based categories to change the enrichment. When D G = F G ,the morphism categories are discrete (identity morphisms only). Since the classify-ing space of a discrete category is isomorphic to itself, we can identify F topG with F G . It follows that D topG is a category of operators over F G in the sense of [39, § Notation 9.3.
Since the category G U ∗ is closed monoidal, it is enriched over itself.We denote this enriched category by G U ∗ . Its based G -spaces of morphisms aregiven by the spaces of all nonequivariant based maps, based at the constant maps atthe basepoint, with G acting by conjugation. We denote by D topG - G U ∗ the categoryof G U ∗ -enriched functors X : D topG −→ G U ∗ . The enrichment over based G -spacesimplies that X (0) = ∗ [39, Lemma 1.13]. To emphasize that these are just (enriched)functors to G -spaces, we call them D topG - G -spaces. In particular, an F G - G -space will mean an object of F G - G U ∗ .Recall that we write D G - Alg as shorthand for the category of D G -algebras andstrict maps in Cat ( G U ∗ ). Proposition 9.4.
Applying the classifying space functor levelwise induces a functor B : D G - Alg −→ D topG - G U ∗ . Proof.
By Proposition 9.1 the classifying space functor B is lax symmetric monoidal.It follows formally that it induces a map on functor categories. Explicitly, if X is a D G -algebra in Cat ( G U ∗ ), we obtain the D topG - G -space B X by applying B levelwise,with action maps given by the composites B D G ( m α , n β ) ∧ B X ( m α ) −→ B ( D G ( m α , n β ) ∧ X ( m α )) Bθ −−→ B X ( n β ) , ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 63 where the first map is the monoidal constraint for B . The commutativity of thecomposition and unit diagrams follows from their analogs for X (see Definition 4.18)and the axioms for a lax monoidal functor. The functoriality of B on strict algebramaps is obtained by applying B levelwise. (cid:3) We now concentrate on the case of F G . The categories F G - G U ∗ and F G - Alg are symmetric monoidal via Day convolution.
Proposition 9.5.
The functor B : F G - Alg −→ F G - G U ∗ of Proposition 9.4 is lax symmetric monoidal.Proof. This follows formally from Proposition 9.1, but we sketch the argument. Theinduced functor B on our categories of algebras preserves the monoidal unit, whichin both the source and the target is given by the representable functor F G ( , − ).Given X and Y in F - Alg , we construct a map B X ∧ B Y −→ B ( X ∧ Y )in F G - G U ∗ by applying the universal property of Day convolution to the map of( F G ∧ F G )- G -spaces with components given by the composites B X ( m α ) ∧ B Y ( n β ) −→ B ( X ( m α ) ∧ Y ( n β )) −→ B ( X ∧ Y )( mn α ⊗ β ) . Here the first map is the lax monoidal constraint for B and the second map isobtained by applying B to the components of the unit of the Day convolutionadjunction. It is routine to check that this map satisfies the required compatibilitieswith the unit, associativity and symmetry isomorphisms. (cid:3) Remark 9.6.
Note that if D G is a Cat ( G U ∗ )-category of operators over F G equipped with a pseudo-commutative structure (Definition 6.13), this does not giverise to a symmetric monoidal structure on D topG . As a result, we do not have amonoidal structure on the category of D topG -algebras, and so we cannot expect ananalogue of Proposition 9.5 for D G -algebras.9.2. From F G - G -spaces to G -spectra. In this section, we first recall the prop-erties of the equivariant Segal machine, whose construction is given in detail in [39].A treatment that deals with multiplicative properties can be found in [15]. In thispaper, we treat the Segal machine as a black box, and we refer the reader to thosesources for details.All homotopical versions of the Segal machine come in the form of bar con-structions, which are only homotopically well-behaved when the input functors X : F G −→ G U ∗ take values in nondegenerately based G -spaces. However, inthe previous subsection, we concentrated on formal properties of our constructions.Write G T and T G for the full subcategories of nondegenerately based G -spaces in G U ∗ and in G U ∗ . Since these categories are not bicomplete, they are less usefulfor formal purposes. We introduce notations and definitions to help deal with theresulting dichotomy. Notation 9.7.
Let D topG be a G U ∗ -category of operators over F G , such as the onein Definition 9.2. A D topG - G -space is levelwise nondegenerately based if each X ( n α )is nondegenerately based. We write D topG - G T for the full subcategory of D topG - G U ∗ whose objects are levelwise nondegenerately based. In particular, starting with thecommutativity operad, whose terms are one-point G -spaces, this defines the fullsubcategory F G - G T of F G - G U ∗ . Definition 9.8.
Define T to be the composite functor T = St F G ◦ ζ ∗ G ◦ R G : O - Alg ps −→ F G - Alg . Then define O - Alg G T ps to be the full subcategory of O - Alg ps consisting of those O -algebras A such that B T A is in F G - G T . Thus, by definition, the composite B T restricts to a functor O - Alg G T ps −→ F G - G T .The functor T collates the categorical functors studied in previous sections, B passes from categorical data to space level data, and the Segal machine passes fromthere to spectra. That machine will be well-behaved when we restrict it to F G - G T ,and O - Alg G T ps specifies those O algebras that feed into F G - G T . We will show inthe next subsection that most O -algebras of interest are in O - Alg G T ps .To define the notion of a Segal machine S G , we need the key notion of a special F G - G -space . To give a conceptual setting for this notion, observe first that, justas we had on categories, we have a composite functor R topG = P top R top from O top -algebras in G T to D topG - G -spaces, where O top is a operad in G U with associatedcategory of operators D topG . We specialize this to the initial operad O top , whichhas O top (0) = O top (1) = ∗ and all other O top ( j ) = ∅ . Its associated category ofoperators is Π G . Applying R topG to this operad, we obtain a functor R topG : G T −→ Π G - G T .For a based G -space X , R topG ( X ) sends n α to the G -space X n α = G T ( n α , X ).More explicitly, this is X n with G -action given by(9.9) g ( a , . . . , a n ) = ( ga α ( g − )(1) , . . . , ga α ( g − )( n ) ) . The functor R topG is right adjoint to the functor L topG that evaluates a Π G - G -spaceat . For a Π G - G -space Y , the unit δ : Y −→ R topG ( Y ( ))of the adjunction is a map of Π G - G -spaces called the Segal map . At level n α , it isinduced by the n projections n α → [39, Definition 2.28]. Definition 9.10.
We say that a Π G - G -space Y is special if δ is a levelwise weak G -homotopy equivalence. We say that a D topG - G -space, and in particular an F G - G -space, is special if its underlying Π G - G -space is special.Recall that an orthogonal G -spectrum E is a positive Ω- G -spectrum if its adjointstructure maps E V −→ Ω W E V ⊕ W are weak G -equivalences when V G = 0 and is connective if the negative homotopygroups of its fixed point spectra are all zero. ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 65 Definition 9.11. A Segal machine is a functor S G : F G - G U ∗ −→ Sp G togetherwith a natural map of G -spaces ν : X ( ) −→ ( S G X ) such that the following properties hold when X is a special F - G -space in F G - G T .(i) S G X is a connective positive Ω- G -spectrum.(ii) The composite of ν with the adjoint structure map( S G X ) −→ Ω V ( S G X ) V is a group completion for all V such that V G = 0. Remark 9.12.
It is equivalent to replace general V by V = R in (ii).The notion of a group completion of a Hopf G -space is defined as a group com-pletion on all fixed point maps (see [13, Definition 1.9]). The nonequivariant con-struction of the Segal machine was introduced in [47]. The equivariant constructionis due to Shimakawa [48], who started from an unpublished version that is also dueto Segal. It is given a self-contained modernized treatment in [39]. A multiplicativeversion is given in [15]. We refer the reader to those sources for details.From now on, we set S G to be the Segal machine from [39], which is lax monoidalby [15]. We could just as well use the equivalent symmetric monoidal versionfrom [15], but that would not be of any benefit since we lost symmetry with themultifunctor St F G . Moreover, using the machine from [39] will be convenient inSection 10, where we will use several results from [39]. We repeat that we mostlytreat the Segal machine S G as a black box. The only detail from [39] that we willneed to make explicit is a partial description of the construction that allows us todefine the natural map ν : X (1) −→ ( S G X ) required in Definition 9.11. That willbe given where it is used in Section 10.We now restate and prove Theorem A. Theorem 9.13.
Let O be a chaotic E ∞ G -operad in Cat ( G U ) . The functor (9.14) K G = S G ◦ B ◦ St F G ◦ ζ ∗ G ◦ R G : O - Alg ps −→ Sp G . from (1.2) extends to a multifunctor K G : Mult ( O ) −→ Mult ( Sp G ) . For an O -algebra A ∈ O - Alg G T ps , K G A is a connective positive Ω - G -spectrum witha group completion map B A −→ Ω V ( K G A ) V for all V such that V G = 0 .Proof. By Corollary 6.25, Theorem 7.13, Theorem 8.12, Proposition 9.5, and [15,Section 5.2], K G is a composition of multifunctors and is thus a multifunctor.When A is in O - Alg G T ps , B T A is in F G - G T , and we claim that it is special.That will imply the second statement. Since T A is level G -equivalent to R G A ,by Theorem 8.1, the claim follows from the fact that B takes equivalences of G -categories to homotopy equivalences of G -spaces and commutes with R G , in thesense that B R G ∼ = R topG B . (cid:3) The identification of objects in O -Alg G T ps . When the operad O and an O -algebra A are topologically discrete, in the sense that they are categories internal to G Set, A is in O - Alg G T ps since all of our categorical constructions retain discretenessand the geometric realization of a based simplicial set is nondegenerately based. Weshow here that many topologically non-trivial examples, such as those that appearin Section 10, are also in O - Alg G T ps .We require the following definition. Nonequivariantly, its use goes back at leastto Milnor’s classical paper [40], and it was studied in more detail by Dyer andEilenberg [7] and later Lewis [26]. Details of equivariant cofibrations are in [2,Section A.2]. Definition 9.15. A G -space X is G -locally equiconnected ( G -LEC for short) if thediagonal map ∆ : X −→ X × X is a G -cofibration.Examples of G -LEC G -spaces include G -CW-complexes [7, 26]. Every basepointof a G -LEC G -space is nondegenerate [7, Corollary II.8]. The following lemma givessufficient conditions for the classifying space of a chaotic category to be nondegen-erately based. Lemma 9.16.
Suppose that
C ∈
Cat ( G U ∗ ) is chaotic and that Ob C is G -LEC.Then B C has a nondegenerate basepoint.Proof. Since Ob C is G -LEC and C is chaotic, the nerve of C is levelwise G -LEC.Then B C is G -LEC by [26, Corollary 2.4(b)], and in particular it has a nondegen-erate basepoint. (cid:3) We also need the following two general results about G -LEC G -spaces. Lemma 9.17.
Let X be a G -LEC based G -space and n α be a finite based G -set.Then X n α = G U ∗ ( n α , X ) is G -LEC.Proof. The G -space X n α can be viewed as the restriction along the homomorphism G −→ G ≀ Σ n of the G ≀ Σ n -space X n . It then follows from [2, Proposition A.2.6]that X n α is G -LEC. (cid:3) Lemma 9.18.
Let H be a subgroup of G , and let Y be an H -LEC space. Then G × H Y is G -LEC.Proof. The diagonal on G × H Y factors as G × H Y ∆ / / id × ∆ ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ ( G × H Y ) × ( G × H Y ) ∼ = / / ( G × G ) × H × H ( Y × Y ) G × H ( Y × Y ) . ∆ × id ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ The map id × ∆ is the induction from H to G of the H -cofibration ∆ Y , and itfollows that it is a G -cofibration. On the other hand, we claim that the map ∆ × idis the inclusion of a coproduct summand and is therefore a G -cofibration. To seethis, note that the subset { ( g, gh ) | g ∈ G, h ∈ H } ⊂ G × G is a ( G, H × H )-invariant subset, and it is precisely the image under the right ( H × H )-action of ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 67 ∆( G ) ⊂ G × G . Since G is discrete, it follows that we may decompose G × G as a( G, H × H )-equivariant disjoint union of this subset and its complement. Crossingwith Y × Y and passing to ( H × H )-orbits gives a G -equivariant decomposition of( G × G ) × H × H ( Y × Y ) into the image of ∆ × id and its complement. (cid:3) In the remainder of this section, we let O be an operad in Cat ( G U ), and A bean O -algebra. Proposition 9.19. If Ob A is G -LEC and has a disjoint basepoint, then A is in O - Alg G T ps .Proof. We first prove that each object G -space T A ( n α ) of T A is G -LEC. Write W for ζ ∗ G R G A , so that T A = St F G W . The G -space Ob W ( n α ) can be identified with( Ob A ) n α , and is thus G -LEC by Lemma 9.17. Moreover, it has a disjoint basepoint.Recall from Theorem 8.1 that Ob St F G W ( n α ) = _ k β F G ( k β , n α ) ∧ Ob W ( k β ) . Since Ob W ( k β ) is G -LEC, it follows that F G ( k β , n α ) ∧ Ob W ( k β ) is G -LEC. Sincethe basepoint of F G ( k β , n α ) ∧ Ob W ( k β ) is disjoint, the infinite wedge is in factan infinite disjoint union, with an adjoined disjoint basepoint. Since an arbitrarycoproduct of G -LEC G -spaces is again G -LEC, Ob T A ( n α ) is G -LEC.In the proof of Theorem 8.1, we defined a pseudomorphism m : T A = St F G W / / /o/o/o W , each of whose components is an equivalence of categories. Let Y ( n α ) ⊂ T A ( n α )be the subcategory Y ( n α ) = m − n α ( ∗ ). Since m n α is an equivalence of categories, Y ( n α ) is equivalent to the trivial category and is therefore chaotic. The fact that thebasepoint splits off of Ob W ( n α ) implies that Ob Y ( n α ) splits off from Ob T A ( n α )and is therefore G -LEC since Ob T A ( n α ) is G -LEC. Since the basepoint of B T A ( n α )lies in B Y ( n α ) and B Y ( n α ) has a nondegenerate basepoint by Lemma 9.16, thisgives the conclusion. (cid:3) The following example will be used in Theorem 10.1.
Remark 9.20.
We embed G U in Cat ( G U ) by regarding an unbased G -space X asan object of Cat ( G U ) with X as both the object and the morphism G -space andwith the source, target, identity and composition maps all the identity. Similarly,we regard X + as an object of Cat ( G U ∗ ). The free O -algebra generated by X + isthe disjoint union of the categories O ( j ) × Σ j X j with base object ∗ the 0th term. Proposition 9.21. If X ∈ G U is G -LEC and Ob O ( j ) is a (discrete) free Σ j -setfor each j , then O + ( X ) is in O - Alg G T ps .Proof. By Proposition 9.19, it suffices to show that Ob O + ( X ) is G -LEC. This holdsif each G -space Ob (cid:16) O ( j ) × Σ j X j (cid:17) ∼ = Ob O ( j ) × Σ j X j is G -LEC. Since Ob O ( j ) is discrete with free Σ j -action, we can write it as a disjointunion of ( G × Σ j )-sets ( G × Σ j ) / Λ, where Λ ⊂ G × Σ j is a subgroup such that Λ ∩ e × Σ j = e . In other words, the subgroup Λ is the graph of a homomorphism α : H −→ Σ j for some subgroup H ≤ G . For each such Λ, we have an isomorphismof G -spaces (cid:0) ( G × Σ j ) / Λ (cid:1) × Σ j X j ∼ = G × H X j α . Since X is G -LEC, Lemma 9.17 implies that X j α is H -LEC. Then by Lemma 9.18,we have that G × H X j α is G -LEC as wanted. (cid:3) Consider the category of operators D G = D G ( O ). The following analogue ofProposition 9.19, with F G replaced by D G , will be needed in Section 10.2. Therewe use comparisons between infinite loop space machines S G defined on F G - G -spaces and S D G G defined on D topG - G -spaces, where D topG is the topological version of D G , as specified in Definition 9.2. The machine S D G G has good properties when itsdomain is restricted to D topG - G T . Just as for F G , we have categories D G - Alg ps of strict D G -algebras and pseudomorphisms and a subcategory D G - Alg of strict D G -algebras and strict morphisms. Section 8 specializes to give a strictificationfunctor St D G : D G - Alg ps −→ D G - Alg . The composite St D G R G : O - Alg ps −→ D G - Alg plays a role analogous to that of T in the earlier results of this subsection, and we let O - Alg G T ps ( D G ) be the full subcategory of O - Alg ps consisting of those O -algebras A such that B St D G R G A is in D topG - G T . Thus, by definition, B St D G R G restrictsto a functor O - Alg G T ps ( D G ) −→ D topG - G T . Proposition 9.22. If Ob O ( j ) for each j and Ob A are G -LEC, and Ob A has adisjoint basepoint, then A is in O - Alg G T ps ( D G ) .Proof. We need a modification of the first step of the proof of Proposition 9.19 toaccount for strictification over D G rather than F G . Writing Z = R G A , we have Ob St D G Z ( n α ) = _ k β Ob D G ( k β , n α ) ∧ Ob Z ( k β ) . But Ob D G ( k β , n α ) is a finite coproduct of finite products of G -spaces Ob O ( j ),each of which has a disjoint basepoint and is assumed to be G -LEC. Therefore Ob St D G Z ( n α ) is G -LEC. The rest of the proof of Proposition 9.19 goes throughunchanged. (cid:3) The proof of Proposition 9.21 applies directly to give the following analog.
Proposition 9.23. If X ∈ G U is G -LEC and Ob O ( j ) is a (discrete) free Σ j -setfor each j , then O + ( X ) is in O - Alg G T ps ( D G ) . Nondegenerate basepoints and Day convolution.
This brief parentheti-cal section highlights a question that seems to have been overlooked in all previouspapers dealing with the use of Day convolution in topology, even nonequivariantly,whether for spectra or for categories of operators. We concentrate on the latter andrestrict attention to F , thinking nonequivariantly for simplicity. ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 69 Of course, smash products are constructed as quotient spaces X × Y /X ∨ Y in U ∗ . It is essential to be working in compactly generated spaces since otherwisethe smash product is not even associative [36, Theorem 1.7.1]. It follows fromLillig’s union theorem [27] that X ∧ Y is nondegenerately based if X and Y are.Therefore both U ∗ and its full subcategory T are symmetric monoidal under thesmash product. By Remark 2.15, we have associated multicategories Mult ( U ∗ ) and Mult ( T ).An F -space is an (enriched) functor F −→ U ∗ and the category F - U ∗ of F -spaces is symmetric monoidal under the internal smash product given by Day con-volution. By Remark 2.15, it also has an associated multicategory Mult ( F - U ∗ ).That can be defined using either the internal smash product as in Remark 2.15or using the external smash product as in Definition 5.15. These definitions giveisomorphic multicategories by the universal property of Day convolution.Now consider the category F - T of (enriched) functors F −→ T . It has beenasserted in many places, including our own [15], that F - T is symmetric monoidalunder the internal smash product. We do not know whether or not that is true,and we believe that it is not. The external smash product X ⊼ Y : F ∧ F −→ U ∗ offunctors X, Y : F −→ T clearly takes values in T , but it does not follow that theinternal smash product X ∧ Y : F −→ U ∗ takes values in T . That is, we do notbelieve that Day convolution preserves levelwise nondegeneracy of basepoints. Wecannot use the universal property to prove that it does, and we have not succeededin proving that it does by direct inspection of the construction.That problem does not affect applications since, using the external smash product as in Definition 5.15, we have the full submulticategory Mult ( F - T ) of Mult ( F - U ∗ ),whose objects are levelwise nondegenerately based functors. When we reinterpret internally , using Day convolution, we may leave that world. The same holds forthe functors from F or, equivariantly, F G to topological G -categories that are thefocus of this paper.9.5. From O -transformations to homotopies of maps of G -spectra. It isclassical that the classifying space functor takes G -categories, G -functors and G -natural transformations to G -spaces, G -maps, and G -homotopies. For the last, G -natural transformations are functors C × I −→ D , where I is the categorywith two objects [0] and [1] and one non-identity morphism [0] −→ [1]. For based G -categories, based G -transformations are given by based G -functors C ∧ I + −→ D .Recall the definition of O -transformations from Definition 2.12. Proposition 9.24.
The functor K G takes O -transformations to homotopies ofmaps of G -spectra.Proof. We showed in [15, Proposition 6.16] that the topological Segal machine S G preserves homotopies. If we start with F G -algebras in Cat ( G U ∗ ), which of course are themselves G -functors, then maps between them are G -natural trans-formations and maps between those are G -modifications. These are given level-wise by G -categories, G -functors, and G -natural transformations. Since B com-mutes with products, it takes F G - G -algebras in Cat ( G U ∗ ), F G -functors, and F G -transformations between them to G U ∗ -enriched functors F G −→ G U ∗ , enrichednatural transformations, and homotopies between those. As R G , ζ ∗ G , and St F G are all 2-functors, with St F G converting pseudostructure to strict structure, theircomposite takes O -transformations to F G -transformations, which are levelwise G -natural transformations. (cid:3) The result above is used in [12, Remark 2.9], but it will surely find other uses.10.
The multiplicative Barratt-Priddy-Quillen Theorem
In this section, we prove Theorem B. We begin by producing the transformation α in Section 10.1. We show that α X is a stable equivalence of orthogonal G -spectrafor all G -LEC G -spaces X in Section 10.2 and we finish by showing that α ismonoidal in Section 10.3.10.1. The construction of α . We restate and begin the proof of Theorem B.
Theorem 10.1.
Let O be a topologically discrete chaotic E ∞ G -operad in Cat ( G U ) .Then there is a lax monoidal natural transformation α : Σ ∞ G + −→ K G O + of functors G U −→ Sp G such that α X is a stable equivalence of orthogonal G -spectrafor all G -LEC G -spaces X . Recall that we write T for the composite T = St F G ◦ ζ ∗ G ◦ R G : O - Alg ps −→ F G - Alg , so that K G is given by K G = S G ◦ B ◦ T . We shall exploit the fact that Σ ∞ G + : G U −→ Sp G is left adjoint to the zeroth G -space functor ( − ) , with basepoint forgotten, toconstruct α . Therefore, to define α : Σ ∞ G + −→ K G O + in Theorem 10.1, it sufficesto define a map of unbased G -spaces˜ α X : X −→ S G (cid:0) B TO + X (cid:1) for each unbased G -space X . We define ˜ α X to be the composite displayed in thediagram X ∼ = (cid:15) (cid:15) ˜ α X / / S G (cid:0) B TO + X (cid:1) BX Bη (cid:15) (cid:15) B O + X (cid:0) B R G O + X (cid:1) ( ) Bi / / (cid:0) B TO + X (cid:1) ( ) . ν O O ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 71 Regarding X as an object of Cat ( G U ) as in Remark 9.20, the top left isomorphismis immediate; η on the left is the unit of the monad O + . For the bottom left equality,it is true by definition that A = ( R G A )( ) for any O -algebra A in Cat ( G U ), suchas A = O + X . Next, for any strict D G -algebra Y , such as Y = R G O + X , the map i : Y ( ) −→ St F G ζ ∗ G Y ( )is given by Y ( ) = ζ ∗ G Y ( ) i −→ St F G ζ ∗ G Y ( ) , where i is a component of the unit of the adjunction of Theorem 8.1 and is anequivalence. Finally, the map ν is the natural map required by Definition 9.11; itwill be specified in Section 10.2.The adjoints of the maps ˜ α X define the natural transformation α , and we mustverify that α is monoidal and homotopical, the latter meaning that α is a stable G -equivalence.10.2. The proof that α is a stable equivalence. We will show that α X is anequivalence by comparing it with the equivariant Barratt-Priddy-Quillen equiva-lence for the equivariant operadic machine proven in [13] and the equivalence be-tween the equivariant operadic and Segal machines proven in [39]. Consider thefollowing diagram of G -spectra, in which O top + denotes the monad on unbased G -spaces associated to the operad B O , E G denotes the operadic infinite loop spacemachine of [13, Definition 2.7], E D G G denotes the category of operators infinite loopspace machine of [39, Definition 6.19], and S D G G denotes the Segal machine on D topG - G -spaces of [39, § ∞ G X + (cid:15) (cid:15) α X / / S G B TO + X E G O top + X ∼ = (cid:15) (cid:15) S D G G B R G O + X O O ✤✤✤ E D G G R topG O top + X / / ❴❴❴❴❴❴ S D G G R topG O top + X ∼ = O O The arrows 1 through 5 are specified as follows.(1) The map 1 is the stable equivalence of orthogonal G -spectra given in [13,Theorem 6.1], which is an operadic version of the Barratt-Priddy-QuillenTheorem.(2) The isomorphism of 2 is a comparison between operad level and categoryof operators level infinite loop space machines given by [39, Corollary 6.22]. We repeat that numbered references to [39] refer to the first version posted on ArXiv; theywill be changed when the published version appears. (3) The dashed arrow 3 is a zig-zag of stable equivalences between the gener-alized operadic and Segal machines, both defined on D topG - G -spaces. Thisis given by [39, Theorem 7.1](4) The isomorphism 4 is induced by the isomorphism R topG B ∼ = B R G .(5) The zig-zag of level equivalences 5 is described in (10.3) below.Write Y for the D G -algebra R G O + X . Theorem 8.1, Corollary 7.10, and Lemma 8.6give a zig-zag of strict maps of D G -algebras in Cat ( G U ∗ ) that are level equivalences Y St D G Y ∼ m o o ∼ St D G χ / / St D G ξ ∗ G ζ ∗ G Y ∼ ψ / / ξ ∗ G St F G ζ ∗ G Y . We apply B to this and use that ξ ∗ G commutes with B to obtain a zigzag of levelequivalences of D topG - G -spaces. B Y B St D G Y ∼ Bm o o ∼ Bψ ◦ B St D G χ / / ξ ∗ G B St F G ζ ∗ G Y . The D topG - G -space B Y has levelwise disjoint basepoints, whereas the basepoints of B St D G Y and ξ ∗ G B St F G ζ ∗ G Y are levelwise nondegenerate according to Propositions9.23 and 9.21, respectively. By [39, Proposition 4.31], the Segal machine S D G G on D topG - G -spaces converts this to a zig-zag of stable equivalences of orthogonal G -spectra. By [39, Theorem 4.32], we have a natural stable equivalence S D G G ξ ∗ G −→ S G relating the Segal machine on D topG - G -spaces to the Segal machine on F G - G -spaces. Applying this to the last term in the sequence above and rememberingthat T = St F G ζ ∗ G R G and Y = R G O + X , we obtain the final zig-zag 5 of stableequivalences of orthogonal G -spectra:(10.3) S D G G B R G O + X S D G G B St D G R G O + X ∼ / / ∼ o o S D G G ξ ∗ G B TO + X ∼ / / S G B TO + X. In order to deduce that α X is a stable equivalence, we need only show that (10.2)yields a commutative diagram in the homotopy category. As we are mapping outof a suspension spectrum, we may instead consider the diagram on zeroth spaces,by adjunction. Chasing the required diagrams is tedious but routine. We givea few details for the skeptical reader. For this, we only need to know that theoperadic and Segal machines are given by two-sided monadic and categorical barconstructions with easily described zeroth spaces and maps between them. We usethe notations from [39], and we refer the reader to that source for more detailsof the definitions of the constructions and maps between them. Abbreviating bywriting q = Bψ ◦ B St D G χ , the adjoint of (10.2) is a diagram of unbased G -spacesthat takes the following form: ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 73 (10.4) X (cid:15) (cid:15) Bη / / + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ (cid:16) (cid:16) ✾ ✼ ✺ ✸ ✶ ✴ ✳ ✱ ✯ ✮ ✬ ✫ ✪★✧ ' ' PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP ( B R G O + X )( ) Bi / / ( B St F G ζ ∗ G R G O + X )( ) ν / / B (( S ) • , F G , B TO + X ) B (( S ) • , D topG , ξ ∗ G B TO + X ) ∼ O O O top + X ∼ = (cid:15) (cid:15) B (( S ) • , D topG , B St D G R G O + X ) ∼ q O O ∼ Bm (cid:15) (cid:15) B (( S ) • , D topG , B R G O + X ) B (( S ) • , D topG , R G O top + X ) ∼ = O O ι (cid:15) (cid:15) O top + X B ( • ( S ) , D topG , R G O top + X ) ω o o / / B (( S ) • , D topG , R G O top + X ) ι / / B ( I + ∧ ( S ) • , D topG , R G O top + X ) . Here the numbers n labelling solid arrows are the induced maps on 0th G -spacesfrom the maps n in (10.2). The zig-zag of morphisms starting from the bottomleft of the diagram and ending at the source of 4 is the zig-zag of maps of 0th G -spaces induced by 3 in (10.2), while the top three vertical maps on the rightare the zig-zag induced by 5 . The top horizontal composite is the adjoint of α X .In the middle left entry, we use an identification for the operadic machine that isexplained in [13, Remark 2.9]; the map 1 is just η top : X −→ O top + X ∼ = B O + X , theisomorphism that of (3.19). At the top, ( B R G O + X )( ) = B O + X , and Bη = η top .We will define the dotted arrow maps 6 , 7 , and 8 and show that each subdia-gram commutes, at least up to homotopy. The maps 6 , 7 , and 8 each map intothe summand labeled by in the space of 0-simplices of its corresponding (categori-cal) bar construction. These two-sided bar constructions are of the form B ( Y, E , Z ),where E is a G U ∗ -enriched category, and Y : E op −→ G U ∗ and Z : E −→ G U ∗ are G U ∗ -enriched functors. The space of 0-simplices is given by _ n Y ( n ) ∧ Z ( n ) , where n ranges over the objects of E . In all but one of the bar constructions in thediagram (the bottom right corner), Y ( ) = S , hence the summand Y ( ) ∧ Z ( ) isisomorphic to Z ( ). Taking E = F G , the Segal machine S G Z is constructed as( S G Z ) V = B (( S V ) • , F G , Z ) . Taking V = 0, the map ν : Z ( ) −→ ( S G Z ) is given by the inclusion of Z ( ) inthe space of 0-simplices of the bar construction. The map ω in (10.4) is given byprojection to the relevant component, as in [39, Section 7.6]. We here use a choice of bar construction that is slightly modified from but equivalent to thechoice in the ArXiv version of [39]; it will appear in the published version and in a revision to beposted on ArXiv. It is the choice labelled B N G in [15]. Replacing bar constructions in (10.4) by the components of their zero simplicesthat serve as targets for the maps with domain X , the diagram can be written as(10.5) X η (cid:15) (cid:15) Bη / / * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ (cid:16) (cid:16) ✵ ✳ ✲ ✱ ✰ ✯ ✮ ✭ ✬ ✫ ✪ ✩ ★✧✦ $ $ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■ B O + X Bi / / B TO + X ( ) ν B TO + X ( )D ξ ∗ G B TO + X ( ) O top + X C B St D G R G O + X ( ) ∼ q O O ∼ Bm (cid:15) (cid:15) B R G O + X ( )A B O top + X ∼ = O O ι (cid:15) (cid:15) O top + X O top + X ω O top + X ι / / I + ∧ O top + X We take 6 and 7 both to be η , making diagram A commute and diagram Bcommute up to homotopy. We take 8 to be the composite X Bη −−→ B O + X = B R G O + X ( ) Bi D −−−→ B St D G R G O + X ( ) . Region C of the diagram commutes by the triangle identity, since i and m arethe unit and counit for the adjunction in Theorem 8.1. Region D is obtained byapplying the classifying space functor B to the following diagram. X η (cid:15) (cid:15) η / / O + X i F / / TO + X ( ) R G O + X ( ) i D (cid:15) (cid:15) χ / / ξ ∗ G ζ ∗ G R G O + X ( ) ξ ∗ G i F / / i D (cid:15) (cid:15) ξ ∗ G St F G ζ ∗ G R G O + X ( )St D G R G O + X ( ) St D G χ / / St D G ξ ∗ G ζ ∗ G R G O + X ( ) ψ ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ The top rectangle commutes because the components of χ (Corollary 7.10) areidentity maps. The lower rectangle commutes by the naturality of i , and the trianglecommutes by Lemma 8.6.10.3. The proof that α is monoidal. Since the adjoint of e α {∗} is easily seen tobe the unit map of the lax monoidal functor K G O + , it suffices to verify that thefollowing diagram commutes for G -spaces X and Y . Recall again that ( X × Y ) + ∼ = ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 75 X + ∧ Y + . Σ ∞ G X + ∧ Σ ∞ G Y + α ∧ α / / ∼ = (cid:15) (cid:15) K G O + X ∧ K G O + Y ϕ (cid:15) (cid:15) Σ ∞ G ( X × Y ) + α / / K G O + ( X × Y )The map ϕ is constructed by applying the composite of the multifunctors fromTheorems 3.28 and 9.14 to the identity map X × Y −→ X × Y considered as a2-ary morphism in Mult ( G U ).By passage to adjoints, and since the adjunction is monoidal, we are reduced toshowing that the following diagram of maps of G -spaces commutes. X × Y ˜ α X × ˜ α Y / / = (cid:15) (cid:15) ( K G O + X ) × ( K G O + Y ) (cid:15) (cid:15) ( K G O + X ∧ K G O + Y ) ϕ (cid:15) (cid:15) X × Y ˜ α X × Y / / ( K G O + ( X × Y )) Here, the top right map is the lax monoidal constraint for the zeroth space functor.The proof is concluded by checking that the following diagram commutes. X × Y Bη × Bη / / Bη (cid:15) (cid:15) B O + X × B O + Y Bi × Bi / / B TO + X ( ) × B TO + Y ( ) ν × ν (cid:15) (cid:15) B O + ( X × Y ) Bi (cid:15) (cid:15) ( S G B TO + X ) × ( S G B TO + Y ) (cid:15) (cid:15) B TO + ( X × Y )( ) ν / / ( S G B TO + ( X × Y )) (cid:0) ( S G B TO + X ) ∧ ( S G B TO + Y ) (cid:1) ϕ o o Coherence axioms
We collect the coherence axioms we need in this section. Those for pseudo-commutative operads, deferred from Definition 3.10, appear in Section 11.1; thosefor
Mult ( O ), deferred from Definition 3.14, appear in Section 11.2; and those for Mult ( D ), deferred from the unpacking of Definition 5.15 in Section 5.3, are gath-ered in Section 11.3.11.1. Coherence axioms for pseudo-commutative operads.
Here we com-plete Definition 3.10 by specifying coherence axioms for the α j,k of diagram (3.11).(i) The component of α ,n at ( , y ) is the identity map. (ii) The composite O ( j ) × O ( k ) ⊛ / / t (cid:15) (cid:15) ✌✌✌✌ (cid:2) (cid:10) α j,k O ( jk ) τ k,j (cid:15) (cid:15) O ( k ) × O ( j ) ⊛ / / t (cid:15) (cid:15) ✌✌✌✌ (cid:2) (cid:10) α k,j O ( kj ) τ j,k (cid:15) (cid:15) O ( j ) × O ( k ) ⊛ / / O ( jk ) . is the identity V -transformation.(iii) For permutations ρ ∈ Σ k and ρ ∈ Σ j , O ( j ) × O ( k ) ⊛ / / t (cid:15) (cid:15) ✌✌✌✌ (cid:2) (cid:10) α j,k O ( jk ) τ k,j (cid:15) (cid:15) O ( k ) × O ( j ) ⊛ / / ρ × ρ (cid:15) (cid:15) O ( kj ) ρ ⊗ ρ (cid:15) (cid:15) O ( j ) × O ( k ) ⊛ / / O ( jk ) . = O ( j ) × O ( k ) ⊛ / / ρ × ρ (cid:15) (cid:15) O ( jk ) ρ ⊗ ρ (cid:15) (cid:15) O ( j ) × O ( k ) ⊛ / / t (cid:15) (cid:15) ✌✌✌✌ (cid:2) (cid:10) α j,k O ( jk ) τ k,j (cid:15) (cid:15) O ( k ) × O ( j ) ⊛ / / O ( kj ) . (iv) Let j Y i =1 O ( k i ) × O ( ℓ ) ∆ ℓ −−→ j Y i =1 (cid:16) O ( k i ) × O ( ℓ ) (cid:17) and O ( ℓ ) × j Y i =1 O ( k i ) ∆ ′ ℓ −−→ j Y i =1 (cid:16) O ( ℓ ) × O ( k i ) (cid:17) ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 77 be the morphisms whose i th components are the products p i × id and id × p i ,respectively. We require the 2-cell O ( j ) × j Q i =1 O ( k i ) × O ( ℓ ) ∼ = (cid:15) (cid:15) id × ∆ ℓ / / O ( j ) × j Q i =1 (cid:16) O ( k i ) × O ( ℓ ) (cid:17) id × Q ⊛ / / ∼ = (cid:15) (cid:15) ✒✒✒✒ (cid:5) (cid:13) id × Q α ki,ℓ O ( j ) × j Q i =1 O ( k i ℓ ) id × Q τ ℓ,ki (cid:15) (cid:15) O ( j ) × O ( ℓ ) × j Q j =1 O ( k i ) id × Q ∆ ℓ (cid:15) (cid:15) id × ∆ ′ ℓ / / O ( j ) × j Q j =1 (cid:16) O ( ℓ ) × O ( k i ) (cid:17) id × Q ⊛ / / O ( j ) × j Q i =1 O ( ℓk i ) γ (cid:15) (cid:15) O ( j ) × O ( ℓ ) × j Q j =1 O ( k i ) ℓ ⊛ × id / / ∼ = (cid:15) (cid:15) ✓✓✓✓ (cid:5) (cid:13) α j,ℓ × id O ( jℓ ) × j Q i =1 O ( k i ) ℓ γ / / τ ℓ,j × id (cid:15) (cid:15) O ( ℓk ) D ℓ,k ∗ (cid:15) (cid:15) O ( ℓ ) × O ( j ) × j Q i =1 O ( k i ) ℓ ⊛ × id / / ∼ = (cid:15) (cid:15) O ( ℓj ) × j Q i =1 O ( k i ) ℓ ∼ = (cid:15) (cid:15) O ( ℓ ) × O ( j ) × (cid:16) j Q i =1 O ( k i ) (cid:17) ℓ ⊛ × id / / O ( ℓj ) × (cid:16) j Q i =1 O ( k i ) (cid:17) ℓ γ / / O ( ℓk ) to be equal to the 2-cell O ( j ) × j Q i =1 O ( k i ) × O ( ℓ ) γ × id (cid:15) (cid:15) id × ∆ ℓ / / O ( j ) × j Q i =1 (cid:16) O ( k i ) × O ( ℓ ) (cid:17) id × Q ⊛ / / O ( j ) × j Q i =1 O ( k i ℓ ) γ (cid:15) (cid:15) O ( k ) × O ( ℓ ) ⊛ / / ∼ = (cid:15) (cid:15) ✚✚✚✚ (cid:9) (cid:17) α k,ℓ O ( kℓ ) τ ℓ,k (cid:15) (cid:15) O ( ℓ ) × O ( k ) ⊛ / / O ( kℓ ) . Here, in the first 2-cell above, D ℓ,k ∗ is the distributivity isomorphism specified inthe following definition. Definition 11.1.
Given ℓ , j , and k , . . . , k j , let k = P k i and define D ℓ,k ∗ : ℓ ⊗ ( k ⊕ · · · ⊕ k j ) −→ ( ℓ ⊗ k ) ⊕ · · · ⊕ ( ℓ ⊗ k j )in the bipermutative category Σ. It is given explicitly as the permutation D ℓ,k ∗ = (cid:0) τ k ,ℓ ⊕ · · · ⊕ τ k j ,ℓ (cid:1) τ ℓ,k . Alternatively, using the operad structure on
Assoc , we can identify D ℓ,k ∗ as D ℓ,k ∗ = γ ( τ ℓ,j ; ∆ ℓ ( e k , . . . , e k j )) . It is the permutation in Σ ℓk that permutes blocks of sizes k , . . . , k j , . . . , k , . . . , k j according to τ ℓ,j . Remark 11.2.
In the language of Corner and Gurski [5, Theorem 4.6], axiom (ii)states that we require pseudo-commutativity structures to be symmetric. Axiom (iii) is an equivariance axiom that is necessary in order for α to induce a map atthe monad level (Proposition 3.22), but which was unfortunately omitted in [5].Axiom (iv) encodes the compatibility of α with operadic composition, and isgiven in [5, Theorem 4.4]. Unpacking axiom (iv), it states that given x ∈ O ( j ), y i ∈ O ( k i ), and z ∈ O ( ℓ ), the following diagram commutes: γ ( x ; y ⊛ z, . . . , y j ⊛ z ) (cid:0) τ ℓ,k ⊕ · · · ⊕ τ ℓ,k j (cid:1) D ℓ,k ∗ γ ( x ; y ⊛ z, . . . , y j ⊛ z ) τ ℓ,k γ (cid:16) x ; ( y ⊛ z ) τ ℓ,k , . . . , ( y j ⊛ z ) τ ℓ,k j (cid:17) D ℓ,k ∗ γ (id; α,...,α ) D ℓ,k ∗ (cid:15) (cid:15) γ ( x ; z ⊛ y , . . . , z ⊛ y j ) D ℓ,k ∗ (cid:16) γ ( x ; y , . . . , y j ) ⊛ z (cid:17) τ ℓ,kα (cid:15) (cid:15) γ ( x ⊛ z ; ∆ ℓ ( y ) , . . . , ∆ ℓ ( y j )) D ℓ,k ∗ γ (( x ⊛ z ) τ ℓ,j ; ∆ ℓ ( y , . . . , y j )) γ ( α ;id ,..., id) (cid:15) (cid:15) γ ( z ⊛ x ; ∆ ℓ ( y , . . . , y j )) z ⊛ γ ( x ; y , . . . , y j ) . This formulation is closer to what is stated in [5]. We can summarize it by theequation ( α x,z ⊛ id y ) ◦ [(id x ⊛ α y,z ) D ℓ,k ∗ ] = α xy,z . This axiom plays the role of a hexagon axiom in our context. (There is a secondaxiom relating α to operadic composition in [5, Theorem 4.4], but the two axiomsare equivalent in the presence of the symmetry axiom (ii).)11.2. Coherence axioms for Mult ( O ) . Here we return to the diagram (3.15),which defines the invertible V -transformations δ i central to Definition 3.14, andcomplete that definition. In axioms (ii) and (iv), we will use the shorthand notation A i,j for the product A i × A i +1 × · · · × A j . The map µ appearing in (iv) and (v)is defined immediately following the axioms. The axioms require certain pastingdiagrams to be equal, and in some cases, it will not be immediately apparent thatthe boundaries are equal; we address that issue after the axioms as well. We use × rather than ∧ throughout since the O ( n ) are unbased and their algebras are definedusing powers rather than smash powers.(i) (Unit Object) The V -transformation δ i (0) is the identity. ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 79 (ii) (Equivariance) For any permutation ρ ∈ Σ n , we require the 2-cell A ,i − × O ( n ) × A ni × A i +1 ,k id × ( ρ × id) × id / / id × (id × ρ ) × id (cid:15) (cid:15) A ,i − × O ( n ) × A ni × A i +1 ,k id × θ ( n ) × id (cid:15) (cid:15) A ,i − × O ( n ) × A ni × A i +1 ,n id × θ ( n ) × id / / s i (cid:15) (cid:15) A ,i − × A i × A i +1 ,kF (cid:15) (cid:15) O ( n ) × ( A ,i − × A i × A i +1 ,k ) n id × F n (cid:15) (cid:15) O ( n ) × B n θ ( n ) / / B ✒✒✒✒ E M δ i ( n ) to be equal to the 2-cell A ,i − × O ( n ) × A ni × A i +1 ,k id × ( ρ × id) × id / / s i (cid:15) (cid:15) A ,i − × O ( n ) × A ni × A i +1 ,k id × θ ( n ) × id / / s i (cid:15) (cid:15) A ,i − × A i × A i +1 ,kF (cid:15) (cid:15) O ( n ) × ( A ,i − × A i × A i +1 ,k ) n ρ × id / / id × F n (cid:15) (cid:15) O ( n ) × ( A ,i − × A i × A i +1 ,k ) n id × F n (cid:15) (cid:15) O ( n ) × B n ρ × id / / O ( n ) × B n θ ( n ) / / B . ✎✎✎✎ C K δ i ( n ) (iii) (Operadic Identity): The component of δ i (1) at an object( a , . . . , a i − , ( , a i ) , a i +1 , . . . , a k )is the identity map, where ∈ O (1) is the unit of the operad O .(iv) (Operadic Composition): We require δ i to be compatible with composition inthe operad. In order to save space, we choose to display only the case of i = k ,but the general case is analogous.The composite 2-cell A ,k − × O ( n ) × Q r (cid:18) O ( m r ) × A m r k (cid:19) s k (cid:15) (cid:15) id × Q r θ ( m r ) / / A ,k − × O ( n ) × A nks k (cid:15) (cid:15) id × θ ( n ) / / A ,k − × A kF (cid:15) (cid:15) O ( n ) × Q r (cid:18) A ,k − × O ( m r ) × A m r k (cid:19) id × Q r s k (cid:15) (cid:15) id × Q r (id × θ ( m r )) / / O ( n ) × ( A ,k − × A k ) n id × F n (cid:15) (cid:15) O ( n ) × Q r (cid:18) O ( m r ) × ( A ,k − × A k ) m r (cid:19) id × Q r (id × F mr ) (cid:15) (cid:15) ✍✍✍✍ C K δ k ( n ) O ( n ) × Q r (cid:18) O ( m r ) × B m r (cid:19) id × Q r θ ( m r ) / / O ( n ) × B n θ ( n ) / / ✞✞✞✞ ? G id × Q r δ k ( m r ) B is equal to the 2-cell A ,k − × O ( n ) × Q r (cid:18) O ( m r ) × A m r k (cid:19) s k (cid:15) (cid:15) id × µ / / A ,k − × O ( m ) × A mks k (cid:15) (cid:15) id × θ ( m ) / / A ,k − × A kF (cid:15) (cid:15) O ( n ) × Q r (cid:18) A ,k − × O ( m r ) × A m r k (cid:19) id × Q r s k (cid:15) (cid:15) O ( n ) × Q r (cid:18) O ( m r ) × ( A ,k − × A k ) m r (cid:19) id × Q r (id × F mr ) (cid:15) (cid:15) µ / / O ( m ) × ( A ,k − × A k ) m id × F m (cid:15) (cid:15) ✍✍✍✍ C K δ k ( m ) O ( n ) × Q r (cid:18) O ( m r ) × B m r (cid:19) µ / / O ( m ) × B m θ ( m ) / / B (v) (Commutation of δ i and δ j ): For i < j , and omitting the inactive variables A h for h = i or j in order to save space, the 2-cell O ( m ) × A mi × O ( n ) × A nj id × θ ( n ) / / s i (cid:15) (cid:15) O ( m ) × A mi × A j θ ( m ) × id / / s i (cid:15) (cid:15) A i × A jF (cid:15) (cid:15) O ( m ) × ( A i × O ( n ) × A nj ) m id × s mj (cid:15) (cid:15) id × (id × θ ( n )) m / / O ( m ) × ( A i × A j ) m id × F m (cid:15) (cid:15) O ( m ) × ( O ( n ) × ( A i × A j ) n ) mµ (cid:15) (cid:15) id × (id × F n ) m / / O ( m ) × ( O ( n ) × B n ) mµ (cid:15) (cid:15) id × θ ( n ) m / / O ( m ) × B m θ ( m ) % % ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑✘✘✘✘ H P id × δ j ( n ) m ☛☛☛☛ A I δ i ( m ) O ( mn ) × Σ mn ( A i × A j ) mn id × F mn / / O ( mn ) × Σ mn B mn θ ( mn ) / / B is equal to the 2-cell obtained by pasting the 2-cell O ( m ) × A mi × O ( n ) × A nj s j + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ s i s s ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ O ( m ) × ( A i × O ( n ) × A nj ) m id × ( s j ) m (cid:15) (cid:15) O ( n ) × ( O ( m ) × A mi × A j ) n id × ( s i ) n (cid:15) (cid:15) O ( m ) × ( O ( n ) × ( A i × A j ) n ) m µ + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ ❴❴❴❴ + α m,n O ( n ) × ( O ( m ) × ( A i × A j ) m ) nµ s s ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ O ( mn ) × Σ mn ( A i × A j ) mn ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 81 to the left of the pasting diagram O ( m ) × A mi × O ( n ) × A nj θ ( m ) × id / / s j (cid:15) (cid:15) A i × O ( n ) × A nj id × θ ( n ) / / s j (cid:15) (cid:15) A i × A jF (cid:15) (cid:15) O ( n ) × ( O ( m ) × A mi × A j ) n id × s ni (cid:15) (cid:15) id × ( θ ( m ) × id) n / / O ( n ) × ( A i × A j ) n id × F n (cid:15) (cid:15) O ( n ) × ( O ( m ) × ( A i × A j ) m ) nµ (cid:15) (cid:15) id × (id × F m ) n / / O ( n ) × ( O ( m ) × B m ) nµ (cid:15) (cid:15) id × θ ( m ) n / / O ( n ) × B n θ ( n ) % % ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏✘✘✘✘ H P id × δ i ( m ) n ✡✡✡✡ A I δ j ( n ) O ( nm ) × Σ nm ( A i × A j ) nm id × F nm / / O ( nm ) × Σ nm B nm θ ( nm ) / / B . These axioms require explanation. They encode the idea that whenever the δ i combine to give two transformations with the same source functor and the sametarget functor, both with target category B , then they are equal. There is animplicit coherence theorem saying that the diagrams we display generate all others.In all of our diagrams, the interior subdiagrams unoccupied by a 2-cell commuteeither by the definition of an operad or by a naturality diagram.Axioms (i) and (ii) give the compatibilities with basepoints and equivariancenecessary for these multimorphisms to give rise to multimorphisms of O -algebras,as defined by Hyland and Power [18]. In (ii), we must check that the source andtarget functors of the two diagrams displayed are equal. The target functors agreetrivially. The source functors agree by the Equivariance Axiom for θ , the naturalityof s i , and the fact that F n ◦ ρ = ρ ◦ F n . Axiom (iii) corresponds to the OperadicIdentity Axiom and requires no explanation.In (iv), we define the map µ to be the map that shuffles the operad variablesto the left and applies the structure map of the operad in those variables. Thesource and target functors of the two diagrams agree by the compatibility axiomsfor O -algebras.In (v), we abuse notation and again write µ for the effect of passing to orbitsfrom the µ used above. Here the target functors of the first and third diagramsagree trivially but their source functors do not; their left vertical composites differ.After pasting the second diagram to the third, the source functors of the firstdiagram and the composite agree. We note that passage to Σ mn -orbits in the seconddiagram is essential, as in Proposition 3.22; without that, the pseudo-commutativityisomorphism α m,n would not mediate between its source and target functors. Remark 11.3.
These axioms imply further compatibilities of the δ i with the unitobject 0. In particular, it follows that the component of δ i ( n ) at an object( a , . . . , ( x ; a i, , . . . , a i,n ) , . . . , a k ) is id if either a j ∈ A j is 0 for some j = i or all coordinates a i,r of the i th object a i ∈ A ni are 0. Moreover, for 1 ≤ r ≤ n , the 2-cell A ,i − × O ( n ) × A n − i × A i +1 ,k id × ( σ r × id) × id / / id × (id × σ r ) × id (cid:15) (cid:15) A ,i − × O ( n − × A n − i × A i +1 ,k id × θ ( n − × id (cid:15) (cid:15) A ,i − × O ( n ) × A ni × A i +1 ,n id × θ ( n ) × id / / s (cid:15) (cid:15) A ,i − × A i × A i +1 ,kF (cid:15) (cid:15) O ( n ) × ( A ,i − × A i × A i +1 ,k ) n id × F n (cid:15) (cid:15) O ( n ) × B n θ ( n ) / / B ✓✓✓✓ E M δ i ( n ) is equal to the 2-cell A ,i − × O ( n ) × A n − i × A i +1 ,k id × ( σ r × id) × id / / s i (cid:15) (cid:15) A ,i − × O ( n − × A n − i × A i +1 ,k id × θ ( n − × id / / s i (cid:15) (cid:15) A ,i − × A i × A i +1 ,kF (cid:15) (cid:15) O ( n ) × ( A ,i − × A i × A i +1 ,k ) n − σ r × id / / id × F n − (cid:15) (cid:15) O ( n − × ( A ,i − × A i × A i +1 ,k ) n − × F n − (cid:15) (cid:15) O ( n ) × B n − σ r × id / / O ( n − × B n − θ ( n − / / B . ✎✎✎✎ C K δ i ( n − Coherence axioms for Mult ( D ) . Here we return to the diagram (5.17),which defines the invertible V ∗ -transformations δ in the k -ary morphisms of Definition 5.15,and give the necessary coherence conditions. The condition on Π in that definitionalready incorporates conditions on basepoints and identity morphisms. These arethe analogues of axioms (i) and (iii) of Section 11.2. We require the following con-dition on composition in D , which is analogous to the operadic composition axiom(iv) there.(Categorical Composition Axiom) We write θ k for the left vertical composite V i D ( m i , n i ) ∧ V i X i ( m i ) t ∼ = / / V i (cid:0) D ( m i , n i ) ∧ X i ( m i ) (cid:1) V i θ / / V i X ( n i )in (5.17). We write C for the composition in D ∧ k V i D ( n i , p i ) ∧ V i D ( m i , n i ) t ∼ = / / V i (cid:0) D ( n i , p i ) ∧ D ( m i , n i ) (cid:1) V i ◦ / / V i D ( m i , p i ) . The right vertical composite V i D ( m i , n i ) ∧ Y ( m ) ⊛ k ∧ id / / D ( m , n ) ∧ Y ( m ) θ / / Y ( n )in (5.17) is the action Θ k of D ∧ k on Y ◦ ⊛ k . ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 83 With these notations, the following pasting diagrams are required to be equal. V i D ( n i , p i ) ∧ V i D ( m i , n i ) ∧ V i X i ( m i ) id ∧ id ∧ F / / id ∧ θ k & & ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ C ∧ id x x ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ☞☞☞☞ (cid:2) (cid:10) id ∧ δ V i D ( n i , p i ) ∧ V i D ( m i , n i ) ∧ Y ( m ) id ∧ Θ k % % ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ V i D ( m i , p i ) ∧ V i X i ( m i ) θ k & & ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ V i D ( n i , p i ) ∧ V i X i ( n i ) id ∧ F / / θ k x x qqqqqqqqqqqqqqqqqqqq ☞☞☞☞ (cid:2) (cid:10) δ V i D ( n i , p i ) ∧ Y ( n ) Θ k y y tttttttttttttttttt V i X i ( p i ) F / / Y ( p ) V i D ( n i , p i ) ∧ V i D ( m i , n i ) ∧ V i X i ( m i ) id ∧ id ∧ F / / C ∧ id x x ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ V i D ( n i , p i ) ∧ V i D ( m i , n i ) ∧ Y ( m ) id ∧ Θ k % % ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ C ∧ id y y sssssssssssssssssss ❴❴❴❴ k s V i D ( m i , p i ) ∧ V i X i ( m i ) θ k & & ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ id ∧ F / / ☞☞☞☞ (cid:2) (cid:10) δ V i D ( m i , p i ) ∧ Y ( m ) Θ k % % ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ V i D ( n i , p i ) ∧ Y ( n ) Θ k y y tttttttttttttttttt V i X i ( p i ) F / / Y ( p ) The unmarked regions in these diagrams commute. For instance, the rhombusin the first diagram commutes because X , . . . , X k are strict D -algebras, and hence X ∧ . . . ∧X k is a strict D ∧ k -algebra as well. The unlabeled V ∗ -transformation inthe rhombus in the second diagram is the constraint for the D ∧ k -pseudoalgebra Y ◦ ⊛ k . More precisely, it is given by the following whiskering of ϑ , which denotesan iterated version of the coherence V ∗ -pseudotransformation from (5.8). V i D ( n i , p i ) ∧ V i D ( m i , n i ) ∧ Y ( m ) id ∧ ⊛ k ∧ id / / t (cid:15) (cid:15) ✎✎✎✎ (cid:3) (cid:11) ϑ V i D ( n i , p i ) ∧ D ( m , n ) ∧ Y ( m ) id ∧ θ / / ⊛ k ∧ id (cid:15) (cid:15) V i D ( n i , p i ) ∧ Y ( n ) ⊛ k ∧ id (cid:15) (cid:15) V i (cid:0) D ( n i , p i ) ∧ D ( m i , n i ) (cid:1) ∧ Y ( m ) V i ◦∧ id (cid:15) (cid:15) D ( n , p ) ∧ D ( m , n ) ∧ Y ( m ) id ∧ θ / / ◦∧ id (cid:15) (cid:15) D ( n , p ) ∧ Y ( n ) θ (cid:15) (cid:15) V i D ( m i , p i ) ∧ Y ( m ) ⊛ k ∧ id / / D ( m , p ) ∧ Y ( m ) θ / / Y ( p ) . Note that the bottom right rectangle commutes because Y is a strict D -algebra.12. The pseudo-commutativity of D ( O )We prove Theorem 5.12 here. Thus let O be a pseudo-commutative operad in Cat ( V ) and D = D ( O ) the associated category of operators, as in Proposition 5.6.We must construct a V ∗ -pseudofunctor ⊛ : D ∧ D / / /o/o/o D and prove that it gives apseudo-commutative structure. For the sake of clarity, we work with × rather than ∧ in this section; the statements about Π build in basepoint conditions that implythat all the constructions descend to the smash product. We break the proof into several parts. Recall that for a morphism φ : m −→ n of F and 1 ≤ j ≤ n , wewrite φ j = | φ − ( j ) | .When restricted to Π, ⊛ must be ∧ . Thus, on objects, m ⊛ p = mp . At thelevel of Hom categories, the map ⊛ : D ( m , n ) × D ( p , q ) −→ D ( mp , nq )sends the summand in the source labeled by φ : m −→ n and ψ : p −→ q to theone labeled by φ ∧ ψ : mp −→ nq in the target. Therein, the V -functor Y ≤ j ≤ n O ( φ j ) × Y ≤ k ≤ q O ( ψ k ) −→ Y ≤ ℓ ≤ nq O (( φ ∧ ψ ) ℓ ) , is such that its projection onto the ℓ th factor is given by first projecting onto O ( φ j ) × O ( φ k ), where ℓ maps to the pair ( j, k ) under the lexicographic ordering of n ∧ q , and then applying the pairing ⊛ of O . The definition makes sense since( φ ∧ ψ ) ℓ = | ( φ ∧ ψ ) − ( ℓ ) | = | φ − ( j ) || ψ − ( k ) | = φ j ψ k . It is immediate from the definition that ⊛ restricts to ∧ on Π (along ι ) and projectsto ∧ on F (via ξ ), as required.To complete the construction of the V ∗ -pseudofunctor ⊛ , we must prove thefollowing result, which is the heart of the proof that D is pseudo-commutative. Proposition 12.1.
The following diagram of V -functors relating ⊛ to compositioncommutes up to an invertible V -transformation ϑ . D ( n , p ) × D ( r , s ) × D ( m , n ) × D ( q , r ) ⊛ × ⊛ / / id × t × id ∼ = (cid:15) (cid:15) ✒✒✒✒ (cid:5) (cid:13) ϑ D ( nr , ps ) × D ( mq , nr ) ◦ (cid:15) (cid:15) D ( n , p ) × D ( m , n ) × D ( r , s ) × D ( q , r ) ◦×◦ (cid:15) (cid:15) D ( m , p ) × D ( q , s ) ⊛ / / D ( mq , ps ) The collection of such V -transformations descends to the smash product and makes ⊛ : D ∧ D / / /o/o/o D into a V ∗ -pseudofunctor.Proof. The essential combinatorial claim is that the pseudo-commutativity isomor-phisms α of O from Definition 3.10 assemble to give the required invertible V -transformations ϑ . This is not obvious since the α give maps that are not obviouslyrelevant to the diagram. The strategy is to express the results of the source andtarget of the diagram in such a way that the invertible V -transformation betweenthem becomes obvious. In the following equations, we will use the associativityand equivariance formulas from the definition of an operad to massage the twocomposites into comparable form.Since Π and F are permutative categories regarded as V -2-categories, the dia-gram clearly commutes when D = Π or D = F . That is, fixing morphisms ψ : n −→ p , ν : r −→ s , φ : m −→ n , and µ : q −→ r ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 85 in F , we have(12.2) ( ψ ∧ ν ) ◦ ( φ ∧ µ ) = ( ψ ◦ φ ) ∧ ( ν ◦ µ ) . Thus, for the summand labeled by our fixed morphisms ψ , ν , φ , and µ in F , theclockwise and counterclockwise directions land in the same summand of the target.It follows that it suffices to restrict the diagram to these summands. Let1 ≤ j ≤ n, ≤ k ≤ p, ≤ h ≤ r, and 1 ≤ i ≤ s. Looking at the definitions of the composition ◦ of D in (5.5) and of ⊛ , we see thatto chase the diagram starting at the top left with Y k O ( ψ k ) × Y i O ( ν i ) × Y j O ( φ j ) × Y h O ( µ h ) , it suffices to consider its projection onto each term of the product in the target.Thus, we fix k and i , project to O ( ψ k ) × O ( ν i ) × Y j ∈ ψ − ( k ) O ( φ j ) × Y h ∈ ν − ( i ) O ( µ h ) , and then chase. Going around both ways, we land in the term O (cid:16)(cid:0) ( ψ ∧ ν ) ◦ ( φ ∧ µ ) (cid:1) ℓ (cid:17) of the target, where ( k, i ) ∈ p ∧ s corresponds to ℓ ∈ ps under lexicographicalordering.Writing in terms of elements to better apply formulas rather than chase largediagrams, let c ∈ O ( ψ k ) , a ∈ O ( ν i ) , d j ∈ O ( φ j ) , and b h ∈ O ( µ h ) , and recall the permutations ρ defined in (5.5). Going clockwise, the tuple ( c, a, Q j d j , Q h b h )gets sent to(12.3) γ (cid:16) c ⊛ a ; Q ( ψ ∧ ν )( j,h )=( k,i ) d j ⊛ b h (cid:17) ρ ( k,i ) ( ψ ∧ ν, φ ∧ µ ) , and going counterclockwise, it gets sent to(12.4) (cid:16) γ ( c ; Q ψ ( j )= k d j ) ρ k ( ψ, φ ) (cid:17) ⊛ (cid:16) γ ( a ; Q ν ( h )= i b h ) ρ i ( ν, µ ) (cid:17) . In what follows, we use the notation ∆ ℓ ( x ) to denote the ℓ -tuple ( x, . . . , x ). Usingthe definition of ⊛ , the associativity from the definition of an operad twice and abbreviating ρ ( k,i ) ( ψ ∧ ν, φ ∧ µ ) = ρ ( k,i ) , we identify the expression (12.3) as follows: γ (cid:16) c ⊛ a ; Q ( ψ ∧ ν )( j,h )=( k,i ) d j ⊛ b h (cid:17) ρ ( k,i ) = γ (cid:16) γ ( c ; ∆ ψk ( a )); Q ( ψ ∧ ν )( j,h )=( k,i ) γ ( d j ; ∆ φj ( b h )) (cid:17) ρ ( k,i ) = γ (cid:16) c ; Q ψ ( j )= k γ (cid:0) a ; Q ν ( h )= i γ ( d j ; ∆ φj ( b h )) (cid:1)(cid:17) ρ ( k,i ) = γ (cid:16) c ; Q ψ ( j )= k γ (cid:0) γ ( a ; ∆ νi ( d j )); Q ν ( h )= i ∆ φj ( b h ) (cid:1)(cid:17) ρ ( k,i ) = γ (cid:16) c ; Q ψ ( j )= k γ (cid:0) a ⊛ d j ; Q ν ( h )= i ∆ φj ( b h ) (cid:1)(cid:17) ρ ( k,i ) . (12.5)We abbreviate ρ k ( ψ, φ ) = ρ k and ρ i ( ν, µ ) = ρ i . Using Remark 3.7 and theassociativity axiom of the operad, we identify the expression (12.4) as follows: (cid:16) γ ( c ; Q ψ ( j )= k d j ) ρ k (cid:17) ⊛ (cid:16) γ ( a ; Q ν ( h )= i b h ) ρ i (cid:17) = (cid:16) γ ( c ; Q ψ ( j )= k d j ) ⊛ γ ( a ; Q ν ( h )= i b h ) (cid:17) ( ρ k ⊗ ρ i )= γ (cid:16) γ ( c ; Q ψ ( j )= k d j ); ∆ ( ψ ◦ φ ) k ( γ ( a ; Q ν ( h )= i b h )) (cid:17) ( ρ k ⊗ ρ i )= γ (cid:16) c ; Q ψ ( j )= k γ ( d j ; ∆ φj ( γ ( a ; Q ν ( h )= i b h ))) (cid:17) ( ρ k ⊗ ρ i )= γ (cid:16) c ; Q ψ ( j )= k γ (cid:0) γ ( d j ; ∆ φj ( a )); ∆ φj ( Q ν ( h )= i b h ) (cid:1)(cid:17) ( ρ k ⊗ ρ i )= γ (cid:16) c ; Q ψ ( j )= k γ (cid:0) d j ⊛ a ; ∆ φj ( Q ν ( h )= i b h ) (cid:1)(cid:17) ( ρ k ⊗ ρ i )(12.6)The similarity between the reinterpretations (12.5) and (12.6) of (12.3) and (12.4)is clear. We use the α ’s from Definition 3.10 to build an invertible V -transformationfrom the expression (12.5) to the expression (12.6). This will specify ϑ . For legibility,we omit the indices on the α ’s. From the pseudo-commutativity of O , for fixed i and j we have an isomorphism α : ( a ⊛ d j ) τ φ j ,ν i −→ d j ⊛ a, ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 87 where τ φ j ,ν i is as in Definition 3.8. It induces the second isomorphism in the com-posite γ (cid:16) a ⊛ d j ; Q ν ( h )= i ∆ φj ( b h ) (cid:17) D φ j ,µ ∗ γ (cid:16) ( a ⊛ d j ) τ φ j ,ν i ; ∆ φj ( Q ν ( h )= i b h ) (cid:17) γ ( α ;∆ φj (id)) (cid:15) (cid:15) γ (cid:16) d j ⊛ a ; ∆ φj ( Q ν ( h )= i b h ) (cid:17) . The first equality is given by the equivariance formula for γ ; the permutation D φ j ,µ ∗ is as in Definition 11.1, with ∗ running through the set ν − ( i ). This is precisely thepermutation of φ j · P ν ( h )= i µ h = φ j · ( ν ◦ µ ) i elements which permutes accordingto τ φ j ,ν i the φ j · ν i blocks of lengths given by the tuple ∆ φj ( Q ν ( h )= i µ h ).By applying γ ( c ; − ) to the product over j ∈ ψ − ( k ) of the composite above, weobtain the second isomorphism in the composite(12.7) γ (cid:16) c ; Q ψ ( j )= k γ (cid:0) a ⊛ d j ; Q ν ( h )= i ∆ φj ( b h ) (cid:1)(cid:17)(cid:16) L ψ ( j )= k D φ j ,µ ∗ (cid:17) γ (cid:16) c ; Q ψ ( j )= k γ (cid:0) a ⊛ d j ; Q ν ( h )= i ∆ φj ( b h ) (cid:1) D φ j ,µ ∗ (cid:17) γ ( id; Q γ ( α ;∆ φj (id)) ) (cid:15) (cid:15) γ (cid:16) c ; Q ψ ( j )= k γ (cid:0) d j ⊛ a ; ∆ φj ( Q ν ( h )= i b h ) (cid:1)(cid:17) . The equality is again given by the equivariance of γ .A straightforward computation, which we omit, gives that ρ ( k,i ) = (cid:16) M ψ ( j )= k D φ j ,µ ∗ (cid:17) · ( ρ k ⊗ ρ i ) . Therefore, multiplying the isomorphism (12.7) by ρ k ⊗ ρ i yields the desired isomor-phism γ (cid:16) c ; Q ψ ( j )= k γ (cid:0) a ⊛ d j ; Q ν ( h )= i ∆ φj ( b h ) (cid:1)(cid:17) ρ ( k,i ) γ ( id; Q γ ( α ;∆ φj (id)) ) ( ρ k ⊗ ρ i ) (cid:15) (cid:15) γ (cid:16) c ; Q ψ ( j )= k γ (cid:0) d j ⊛ a ; ∆ φj ( Q ν ( h )= i b h ) (cid:1)(cid:17) ( ρ k ⊗ ρ i ) from (12.3) to (12.4). Interpreting the proof diagrammatically shows that we havean invertible V -transformation ϑ as needed. Compatibility of ϑ with identity mor-phisms and the fact that it descends to the smash product follow from Lemma 12.10below. Compatibility with composition in D ∧ D is tedious to check, but boils downto repeated use of Axiom (iv) of Definition 3.10. This completes the proof that ⊛ : D ∧ D / / /o/o/o D is a V ∗ -pseudofunctor. (cid:3) Condition (2) of Definition 5.7 holds as a result of the following more generallemma.
Lemma 12.8.
After restricting the domain of the functors in the diagram ofProposition 12.1 to D ( n , p ) × D ( r , s ) × Π( m , n ) × D ( q , r ) −→ D ( mq , ns ) or D ( n , p ) × Π( r , s ) × D ( m , n ) × D ( q , r ) −→ D ( mq , ns ) , the transformation ϑ is the identity.Proof. We use the notation of the construction of ϑ in Proposition 12.1. The firstrestriction is the case when the d j are all ∗ or , and the second restriction is thecase when e is ∗ or . The key is that if either e or d j is ∗ , then α : ( e ⊛ d j ) τ φ j ,ν i −→ d j ⊛ e is the identity map of ∗ . Similarly, by Definition 3.10 (i), if e = , then α is theidentity map of d j , whereas if d j = , then α is the identity map of e . (cid:3) Note that this result in particular implies that as a V ∗ -pseudofunctor, ⊛ restrictsto ∧ on Π. Lemma 12.9.
The V ∗ -pseudofunctor ⊛ is strictly associative in the sense that thefollowing diagram of V ∗ -pseudofunctors commutes. D ∧ ∧ ⊛ / / /o/o/o ⊛ ∧ id (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O D ∧ ⊛ (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O D ∧ ⊛ / / /o/o/o D Proof.
Since this is an equality of V ∗ -pseudofunctors, we need to check equality ofthe level of assignments on objects, V ∗ -functors on morphisms, and pseudofuncto-riality constraints. The equality of assignments on objects follows from the strictassociativity of ∧ in Π. The equality at the level of morphisms follows from thestrict associativity of the pairing of O (Proposition 3.3). For each composite, thepseudofunctoriality constraint is given by a pasting of two instances of the V ∗ -transformation ϑ of Proposition 12.1. As such they are each constructed usinginstances of α and the operadic structure. After some standard simplifications, theequality of these constraints reduces to axiom (iv) of Definition 3.10. (cid:3) ULTIPLICATIVE EQUIVARIANT K -THEORY AND THE BPQ THEOREM 89 Lemma 12.10.
The V ∗ -pseudofunctor ⊛ : D ∧ D / / /o/o/o D has a symmetry V ∗ -pseudo-transformation τ such that the strict monoidal V ∗ - -functors ι : Π −→ D and ξ : D −→ F preserve the symmetry.Proof. The V ∗ -pseudofunctors ⊛ and ⊛ ◦ t : D ∧ D / / /o/o/o D we are comparing havethe same object functions. Given objects m and p , the 1-cell component of τ is given by the permutation τ m,p : mp −→ pm of Definition 3.8, thought of as amorphism in Π ⊂ D . We need invertible V ∗ -transformations D ( m , n ) ∧ D ( p , q ) ⊛ ◦ t / / ⊛ (cid:15) (cid:15) ✒✒✒✒ (cid:5) (cid:13) ˆ τ D ( pm , qn ) ( τ m,p ) ∗ (cid:15) (cid:15) D ( mp , nq ) ( τ n,q ) ∗ / / D ( mp , qn ) . As in the previous proofs, we can restrict to the components of D ( m , n ) and D ( p , q ),which are indexed on morphisms φ : m −→ n and ψ : p −→ q of F . Note that bothmaps send the component of ( φ, ψ ) in the source to that of( ψ ∧ φ ) ◦ τ m,p = τ n,q ◦ ( φ ∧ ψ )in the target (see (3.9)). We thus fix such φ and ψ and start with Q j O ( φ j ) × Q k O ( φ k ), where 1 ≤ j ≤ n and 1 ≤ k ≤ q . Again for simplicity we work withelements c j ∈ O ( φ j ) and d k ∈ O ( ψ k ).Considering permutations as morphisms of Π ⊂ D and using the definition ofcomposition in D , we find that the clockwise composite sends (cid:0) ( c , . . . , c n ) , ( d , . . . , d q ) (cid:1) to Q j,k ( c j ⊛ d k ) τ ψ j ,φ k , and the counterclockwise composite sends it to Q j,k ( d k ⊛ c j ) , with both products ordered in reverse lexicographical order.Applying a product of maps α gives the invertible V ∗ -transformation ˆ τ indi-cated in the diagram. Note that, similar to Lemma 12.8, we have that the 2-cell ˆ τ is the identity when either copy of D is restricted to Π. We leave to thereader the verification of compatibility with composition, and axioms (i) and (ii) ofDefinition 4.30. (cid:3) References [1] Clark Barwick, Saul Glasman, and Jay Shah. Spectral Mackey functors and equivariant alge-braic K-theory, II.
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