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Equivariant Spectra and Mackey Functors

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Equivariant Spectra and Mackey Functors Equivariant spectra and Mackey functors Jonathan Rubin August 15, 2019 Suppose G is a finite group, and let OrbG be the orbit category of G. By Elmendorf's theorem (cf. Talk 2.2), there is an equivalence between the homotopy theory of G-spaces, and the homotopy theory of topological presheaves over OrbG . In this note, we introduce Mackey functors and explain Guillou and May's version of Elmendorf's theorem for G-spectra. This result gives an algebraic perspective on equivariant spectra.1 1 Mackey functors Spectra are higher algebraic analogues of abelian groups. Thus, we begin by con- sidering the fixed points of G-modules, and then we turn to G-spectra in general. Suppose pM, `, 0q is a G-module. By neglect of structure, M is a G-set, and H G therefore its fixed points M – Set pG{H, Mq form a presheaf over OrbG . Spelled out, we have an inclusion MK Ðâ MH for every inclusion K Ă H Ă G of subgroups, ´1 and an isomorphism gp´q : MH Ñ MgHg for every element g P G and subgroup H Ă G. The extra additive structure on M induces extra additive structure on the fixed points of M. Suppose H Ă G is a subgroup. By adjunction, an H-fixed point G{H Ñ M is equivalent to a G-linear map ZrG{Hs Ñ M, and thus MH – homZrGspZrG{Hs, Mq is an abelian group. More interestingly, for any inclusion of subgroups K Ă H Ă G, there is a G-linear map ZrG{Hs Ñ ZrG{Ks that rep- H resents the element H{K rK P ZrG{Ks . Pulling back along it yields an additive \transfer map" ř H K H trK pxq “ rx : M Ñ M . rK H K ¸P { This constellation of data is an example of a Mackey functor. We give an axiomatic definition now, and we reformulate it in conceptual terms later. Definition 1.1. A G-Mackey functor M consists of 1. an abelian group MpHq for every subgroup H Ă G, and 1Many thanks to Mike Hill for his suggestions and comments on this material. 1 2. additive restriction, transfer, and conjugation homomorphisms H H ´1 resK : MpHq Ñ MpKq trK : MpKq Ñ MpHq cg : MpHq Ñ MpgHg q for all subgroups K Ă H Ă G and elements g P G, such that: H H (i) resH “ trH “ id for all subgroups H Ă G, (ii) ch : MpHq Ñ MpHq is the identity map for all subgroups H Ă G and h P H, L H H H L H (iii) resK resL “ resK and trL trK “ trK for all subgroups K Ă L Ă H Ă G, (iv) cg ch “ cgh for all elements g, h P G, 1 1 H gHg ´ H gHg ´ (v) cg resK “ resgKg ´1 cg and cg trK “ trgKg ´1 cg for all subgroups K Ă H Ă G and elements g P G, and (vi) the double coset formula H H L K resL trK “ trLXaKa´1 caresa´1LaXK a L H K P ¸z { holds for all subgroups K, L Ă H Ă G. The double coset formula is an algebraic incarnation of the orbit decomposition H ´1 resL H{K – aPLzH{K L{pL X aKa q. A morphism f : M Ñ N of Mackey functors is a family of group homomorphisms f pHq : MpHq² Ñ NpHq that preserve all restriction, transfer, and conjugation maps. We write MackpGq for the category of all G-Mackey functors. When G is trivial, a G-Mackey functor is the same thing as an abelian group, so it makes sense to ask how ordinary algebraic notions equivariantize. To start, let's consider the integers Z. This is the free abelian group on a single generator. We shall see that the free G-Mackey functor on a single generator (in the G-component) is the Burnside Mackey functor A, described below. Recall that the Grothendieck group or group completion of a commutative monoid M is the initial abelian group receiving an additive map from M. Example 1.2. For any subgroup H Ă G, let ApHq be the Grothendieck group of the commutative monoid of isoclasses of finite H-sets, under disjoint union. Then H H K A is a Mackey functor. Its restrictions are induced by resK : Set Ñ Set , its H K H transfers are induced by indK “ H ˆK p´q : Set Ñ Set , and its conjugations H gHg ´1 are induced by gp´q : Set Ñ Set . When G is trivial, A – Z. It will be convenient to repackage the data in Definition 1.1 as a certain kind of functor. We start by describing a domain category that encodes axioms (i) { (vi). 2 ` ` Definition 1.3. The Lindner category BG is defined as follows. An object of BG ` is a finite G-set. A morphism from X to Y in BG is a span diagram X Ð U Ñ Y of finite G-sets, modulo the equivalence relation that identifies X Ð U Ñ Y with X Ð U1 Ñ Y if there is an isomorphism U – U1 that makes both triangles commute. The composite of X Ð U Ñ Y and Y Ð V Ñ Z is represented by the span X Ð W Ñ Z constructed by pulling back and composing. W U V X Y Z ` The hom sets of BG are commutative monoids. The sum of X Ð U Ñ Y and X Ð V Ñ Y is represented by X Ð U \ V Ñ Y , and the class of X Ð ? Ñ Y is the additive identity. The Burnside category BG is obtained from the Lindner ` category BG by applying group completion homwise. ` Remark 1.4. The category BG is also sometimes called the Burnside category, but we shall reserve this name exclusively for BG . Note that BG is an additive category, meaning it is enriched over the category of abelian groups, and has all finitary biproducts. The empty set ? is the zero object, and the disjoint union X \ Y is the product and coproduct of X and Y in BG . Note also that the category BG is self-dual. We now give a second definition of a Mackey functor. Let Ab denote the category of abelian groups. Definition 1.5. A G-Mackey functor M is a contravariant Ab-enriched functor from the Burnside category BG to the category Ab of abelian groups. Remark 1.6. We could just as well define Mackey functors to be covariant Ab- enriched functors, because BG is self-dual. Note that a Mackey functor automati- cally sends disjoint unions X \ Y of finite G-sets to direct sums because it respects composition and the additive enrichment. op Definitions 1.1 and 1.5 are equivalent. If M : BG Ñ Ab is a Mackey functor in the second sense, then we recover a Mackey functor in the first sense by setting MpHq “ MpG{Hq H id π resK “ MpG{K ÐÝ G{K ÝÑ G{Hq H π id trK “ MpG{H ÐÝ G{K ÝÑ G{Kq ´1 id ´1 p´qg cg “ MpG{gHg ÐÝ G{gHg ÝÑ G{Hq p´qg ´1 id “ MpG{gHg ´1 ÐÝ G{H ÝÑ G{Hq. The double coset formula arises from the pullback squares that define composition in ` op BG , and the values above determine M : BG Ñ Ab by functoriality and additivity. 3 op Example 1.7. The representable Mackey functor BG p´, G{Gq : BG Ñ Ab is isomorphic to A, because there is an equivalence SetG {pG{Hq » SetH that sends a G-map p : X Ñ G{H to its fiber over eH. From here, the Yoneda lemma implies that A is free on a generator in G{G. In general, the representable Mackey functor BG p´, iPI G{Hi q is free on the set of generators txi | i P I u, where xi is regarded as an element in the G{Hi component. ² The category MackpGq is an abelian category. Kernels and cokernels are com- puted pointwise, and similarly for more general small limits and colimits. There is also a tensor product of Mackey functors, given by Day convolution. op Definition 1.8. The box product M ˝N of two Mackey functors M, N : BG Ñ Ab op op is the left Kan extension of b ˝ pM ˆ Nq : BG ˆ BG Ñ Ab along the cartesian product functor ˆ. op op M ˆ N b BG ˆ BG Ab ˆ Ab Ab ˆ M ˝ N “ Lanˆpb ˝ M ˆ Nq op BG By definition, the box product has the following universal property hompM ˝ N, Pq – hompb ˝ pM ˆ Nq, P ˝ ˆq, and for any finite G-set X P BG , the coend formula implies pY ,ZqPBG ˆBG pM ˝ NqpX q – MpY q b NpZq b BG pX , Y ˆ Zq. » The box product makes MackpGq into a closed symmetric monoidal category, in which the Burnside Mackey functor A is the unit. The Mackey functor-valued hom hompM, Nq is defined by an end dual to the coend for M ˝ N. A (commutative) monoid with respect to the box product is called a (commu- tative) Green functor. A Green functor R behaves like a ring to some extent, but there is an asymmetry between its addition and multiplication. Every component RpHq of a Green functor is a ring, but we are not guaranteed any multiplicative H transfer (norm) maps nK : RpKq Ñ RpHq. Roughly speaking, a Tambara functor H is a commutative Green functor equipped with multiplicative norms nK for all sub- groups K Ă H Ă G, which satisfy the double coset formula and a distributive law over the transfers.
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