Mackey Functors Research Articles

The local symbol complex of a reciprocity functor

For a reciprocity functor $\mathcal{M}$ we consider the local symbol complex $\mathcal{M}\otimes^{M}\mathbb{G}_{m}(\eta_{C})\to\oplus_{P\in C}\mathcal{M}(k)\to\mathcal{M}(k)$, where $C$ is a smooth complete curve over an algebraically closed field $k$ with generic point $\eta_{C}$ and $\otimes^{M}$ is the product of Mackey functors. We prove that if $\mathcal{M}$ satisfies certain conditions, then the homology of the above complex is isomorphic to the $K$-group of reciprocity functors $T(\mathcal{M},\underline{CH}_{0}(C)^{0})(Spec k)$.

Biset Functors as Module Mackey Functors and its Relation to Derivators

In this article, we will show that the category of biset functors can be regarded as a reflective monoidal subcategory of the category of Mackey functors on the 2-category of finite groupoids. This reflective subcategory is equivalent to the category of modules over the Burnside functor. As a consequence of the reflectivity, we can associate a biset functor to any derivator on the 2-category of finite categories.

The slice spectral sequence for certain $$RO({C_{p^{n}}} )$$ R O ( C p n ) -graded suspensions of $$H{\underline{\mathbf{Z}}}$$ H Z ̲

We study the slice filtration and associated spectral sequence for a family of \(RO(C_{p^{n}})\)-graded suspensions of the Eilenberg–MacLane spectrum for the constant Mackey functor \(\underline{\mathbb Z}\). Since \(H\underline{\mathbb Z}\) is the zero slice of the sphere spectrum, this begins an analysis of how one can describe the slices of a suspension in terms of the original slices.

Remarks on relative projective of Mackey functors

The aim of this paper is to study and discuss the concept of relative projectivity in the case of Mackey functors. Then we use the definitions of a vertex and of a source of an indecomposable Mackey functor to recall some results of Green's theory. Higman Criterion is the main tool.

The slice spectral sequence for the C4$C_{4}$ analog of real K-theory

Abstract We describe the slice spectral sequence of a 32-periodic C 4 $C_{4}$ -spectrum K [ 2 ] $K_{[2]}$ related to the C 4 $C_{4}$ norm MU ( ( C 4 ) ) = N C 2 C 4 ⁢ MU ℝ ${\mathrm{MU}^{((C_{4}))}=N_{C_{2}}^{C_{4}}\mathrm{MU}_{\mathbb{R}}}$ of the real cobordism spectrum MU ℝ $\mathrm{MU}_{\mathbb{R}}$ . We will give it as a spectral sequence of Mackey functors converging to the graded Mackey functor π ¯ * ⁢ K [ 2 ] $\underline{\pi}_{*}K_{[2]}$ , complete with differentials and exotic extensions in the Mackey functor structure. The slice spectral sequence for the 8-periodic real K-theory spectrum K ℝ $K_{\mathbb{R}}$ was first analyzed by Dugger. The C 8 $C_{8}$ analog of K [ 2 ] $K_{[2]}$ is 256-periodic and detects the Kervaire invariant classes θ j $\theta_{j}$ . A partial analysis of its slice spectral sequence led to the solution to the Kervaire invariant problem, namely the theorem that θ j $\theta_{j}$ does not exist for j ≥ 7 ${j\geq 7}$ .

On equivalences for cohomological Mackey functors

By results of Rognerud, a source algebra equivalence between two p p -blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories of cohomological Mackey functors. We prove this by giving an intrinsic description of cohomological Mackey functors of a block in terms of the source algebras of the block, and then using this description to construct explicit two-sided tilting complexes realising the above mentioned derived equivalence. We show further that a splendid stable equivalence of Morita type between two blocks induces an equivalence between the categories of cohomological Mackey functors which vanish at the trivial group. We observe that the module categories of a block, the category of cohomological Mackey functors, and the category of cohomological Mackey functors which vanish at the trivial group arise in an idempotent recollement. Finally, we extend a result of Tambara on the finitistic dimension of cohomological Mackey functors to blocks.

Partial Tambara structure on the Burnside biset functor, induced from a derivator-like system of adjoint triplets

In the previous article “A Mackey-functor theoretic interpretation of biset functors”, we have constructed the 2-category Sof finite sets with variable finite group actions, in which bicoproducts and bipullbacks exist. As shown in it, biset functors can be regarded as a special class of Mackey functors on S.In this article, we equip S with a system of adjoint triplets, which satisfies properties analogous to a derivator. This system encodes the six operations for finite groups. As a corollary, we show that the associated Burnside rings satisfy analogous properties to a Tambara functor, in the context of biset functor theory.

A Mackey-functor theoretic interpretation of biset functors

In this article, we consider a formulation of biset functors using the 2-category of finite sets with variable finite group actions. We introduce a 2-category S, on which a biset functor can be regarded as a special kind of Mackey functors. This gives an analog of Dress' definition of a Mackey functor, in the context of biset functors.

The extension algebra of some cohomological Mackey functors

Let k be a field of characteristic p. We construct a new inflation functor for cohomological Mackey functors for finite groups over k. Using this inflation functor, we give an explicit presentation of the graded algebra of self-extensions of the simple functor S1G, when p is odd and G is an elementary abelian p-group.

EQUIVARIANT ANDERSON DUALITY AND MACKEY FUNCTOR DUALITY

AbstractWe show that the$\mathbb{Z}$/2-equivariantnth integral MoravaK-theory with reality is self-dual with respect to equivariant Anderson duality. In particular, there is a universal coefficients exact sequence in integral Morava K-theory with reality, and we recover the self-duality of the spectrumKOas a corollary. The study of$\mathbb{Z}$/2-equivariant Anderson duality made in this paper gives a nice interpretation of some symmetries ofRO($\mathbb{Z}$/2)-graded (i.e. bigraded) equivariant cohomology groups in terms of Mackey functor duality.

Correction to “Voevodsky’s motives and Weil reciprocity,” Duke Math. J. 162 (2013), 2751–2796

We correct a mistake in Section 2.4 of the said paper. In [1, Section 2.4], we wrote: “The category Z Span is isomorphic to the full subcategory of Cor consisting of smooth k-schemes of dimension 0.” Tom Bachmann kindly pointed out to us that this statement is incorrect. Here we clarify the relationship between the two categories and show that it does not affect any argument about cohomological Mackey functors (the only Mackey functors appearing in [1]). We retain the notation of [1]. 1 Let Cor0 be the full subcategory of Cor given by 0-dimensional smooth schemes (D etale k-schemes). If f W X ! Y is a surjective morphism of degree d of etale k-schemes, then we have the formula in Cor0, f i f D d: (1) 2 There is a canonical functor W Z Span!Cor0 (2) which is the identity on objects and sends a span (2.1) from [1], X g Z f ! Y; (3) to f i g. DUKE MATHEMATICAL JOURNAL Vol. 164, No. 10, © 2015 DOI 10.1215/00127094-3146068 Received 23 October 2014. Revision received 14 March 2015. 2010 Mathematics Subject Classification. Primary 19E15; Secondary 19A22, 18D10.

On the Grothendieck rings of equivariant fusion categories

In this paper, we describe a Mackey type decomposition for group actions on abelian categories. This allows us to define new Mackey functors which associates to any subgroup the K-theory of the corresponding equivariantized abelian category. In the case of an action by tensor autoequivalences, the Mackey functor at the level of Grothendieck rings has a Green functor structure. As an application we give a description of the Grothendieck rings of equivariantized fusion categories under group actions by tensor autoequivalences on graded fusion categories. In this settings, a new formula for the tensor product of any two simple objects of an equivariantized fusion category is given, simplifying the fusion formula from Burciu and Natale [J. Math. Phys. 54, 013511 (2013)].

Constructing equivariant spectra via categorical Mackey functors

We give a functorial construction of equivariant spectra from a generalized version of Mackey functors in categories. This construction relies on the recent description of the category of equivariant spectra due to Guillou and May. The key element of our construction is a spectrally-enriched functor from a spectrally-enriched version of permutative categories to the category of spectra that is built using an appropriate version of K-theory. As applications of our general construction, we produce a new functorial construction of equivariant Eilenberg--MacLane spectra for Mackey functors and for suspension spectra for finite G-sets.

Symmetric cohomology of groups as a Mackey functor

Symmetric cohomology of groups, defined by M. Staic in [2], is similar to the way one defines the cyclic cohomology for algebras. We show that there is a well-defined restriction, conjugation and transfer map in symmetric cohomology, which form a Mackey functor under a restriction. Some new properties for the symmetric cohomology group using normalized cochains are also given.

Trace maps for Mackey algebras

Let G be a finite group and R be a commutative ring. The Mackey algebra μR(G) shares a lot of properties with the group algebra RG however, there are some differences. For example, the group algebra is a symmetric algebra and this is not always the case for the Mackey algebra. In this paper we present a systematic approach to the question of the symmetry of the Mackey algebra, by producing symmetric associative bilinear forms for the Mackey algebra.Using the fact that the category of Mackey functors is a closed symmetric monoidal category, we prove that the Mackey algebra μR(G) is a symmetric algebra if and only if the family of Burnside algebras (RB(H))H⩽G is a family of symmetric algebras with a compatibility condition.As a corollary, we recover the well known fact that over a field of characteristic zero, the Mackey algebra is always symmetric. Over the ring of integers the Mackey algebra of G is symmetric if and only if the order of G is square free. Finally, if (K,O,k) is a p-modular system for G, we show that the Mackey algebras μO(G) and μk(G) are symmetric if and only if the Sylow p-subgroups of G are of order 1 or p.

Mackey functors and sharpness for fusion systems

We develop the fundamentals of Mackey functors in the setup of fusion systems including an acyclicity condition as well as a parametrization and an explicit description of simple Mackey functors. Using this machinery we extend Dwyer's sharpness results to exotic fusion systems $\mathcal{F}$ on a finite $p$-group $S$ with an abelian subgroup of index $p$.

Subalgebras of the $\mathbb{Z}/2$-equivariant Steenrod algebra

The aim of this paper is to study sub-algebras of the Z/2-equivariant Steenrod algebra (for cohomology with coefficients in the constant Mackey functor F 2) which come from quotient Hopf algebroids of the Z/2-equivariant dual Steenrod algebra. In particular, we study the equivariant counterpart of profile functions, exhibit the equivariant analogues of the classical A(n) and E(n) and show that the Steenrod algebra is free as a module over these.

Equivariant Kasparov theory of finite groups via Mackey functors

Let G be any finite group. In this paper we systematically exploit general homological methods in order to reduce the computation of G -equivariant KK-theory to topological equivariant K-theory. The key observation is that the functor on \mathsf{KK}^G that assigns to a G -C*-algebra A the collection of its K-theory groups \{ K^H_*(A) : H\leqslant G \} admits a lifting to the abelian category of \mathbb{Z}/2 -graded Mackey modules over the representation Green functor for G ; moreover, this lifting is the universal exact homological functor for the resulting relative homological algebra in \mathsf{KK}^G . It follows that there is a spectral sequence abutting to \mathsf{KK}^G_*(A,B) , whose second page displays Ext groups computed in the category of Mackey modules. Due to the nice properties of Mackey functors, we obtain a similar Künneth spectral sequence which computes the equivariant K-theory groups of a tensor product A\otimes B . Both spectral sequences behave nicely if A belongs to the localizing subcategory of \mathsf{KK}^G generated by the algebras C(G/H) for all subgroups H\leqslant G .

Inducing native Mackey functors to biset functors

In this paper, we describe the induction functor from the category of native Mackey functors to the category of biset functors for a finite group G over an algebraically closed field k of characteristic zero. We prove two applications of this description. As the first application, we exhibit that any projective biset functor over k is induced from a (rational) virtual native Mackey functor. The second application is the explicit description of the projective indecomposable biset functors parameterized by simple groups.

Bredon cohomological dimensions for proper actions and Mackey functors

Abstract For groups with a uniform bound on the length of chains of finite subgroups, we study the relationship between the Bredon cohomological dimension for proper actions and the notions of cohomological dimension one obtains by restricting the coefficients of Bredon cohomology to (cohomological) Mackey functors or fixed point functors. We also investigate the closure properties of the class of groups with finite Bredon cohomological dimension for Mackey functors.