# nLab Mackey functor

under construction

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

## Idea

Quite generally, the term Mackey functor refers to an additive functor from a (subcategory of) a category of correspondences (in a disjunctive category $\mathcal{C}$) to possibly any other additive category which however usually is the “base” abelian category. More generally the term now refers to the fairly obvious homotopy theoretic or higher categorical refinements of this concept.

Therefore the concept of Mackey functors is similar to that of sheaves with transfer and as such appears (implicitly) in the discussion of motives (explicitly e.g. in Kahn-Yamazaki 11, section 2, where $\mathcal{C}$ is a category of suitable schemes).

The concept was however introduced and named as such in the context of representation theory (Dress 71) and equivariant homotopy theory/equivariant cohomology (May 96). Here the equivariant homotopy groups $\pi_n(E)$ of a (genuine) G-spectrum $E$ organize into a Mackey functor on $G$-orbits, and this plays a key role in the equivariant Whitehead theorem. In this context of equivariant stable homotopy theory Mackey functors were given a full (∞,1)-category-theoretic formulation in (Barwick 14).

## Definition

We follow the modern account in (Barwick 14).

Let $\mathcal{C}$ be a disjunctive (∞,1)-category and write $Corr_1(\mathcal{C})^\otimes$ for the (∞,1)-category of correspondences in $\mathcal{C}$, regarded as a symmetric monoidal (∞,1)-category with respect to its coproduct (which is a biproduct by disjunctiveness of $\mathcal{C}$).

Write $\mathcal{A} =$Spectra${}^{\oplus}$ for the (∞,1)-category of spectra regarded as a symmetric monoidal (∞,1)-category with respect to direct sum. More generally $\mathcal{A}$ could be any symmetric monoidal stable (∞,1)-category

Then a (spectral) Mackey functor on $\mathcal{C}$ is a monoidal (∞,1)-functor of the form

$S \;\colon\; Corr_1(\mathcal{C})^\otimes \longrightarrow \mathcal{A}^{\oplus} \,.$

Notice that this means that $S$ is in particular

1. a covariant (∞,1)-functor $(-)_\ast \colon\mathcal{C} \to \mathcal{A}$;

2. a contravariant (∞,1)-functor, hence $(-)^\ast \colon\mathcal{C}^{op} \to \mathcal{A}$;

3. satisfying the Beck-Chevalley condition.

(More generally one may specify suitably chosen sub-$(\infty,1)$-categories $\mathcal{C}^\dagger, \mathcal{C}_\dagger \subset \mathcal{C}$ and restrict $Corr_1$ to correspondences whose left leg is in $\mathcal{C}_\dagger$ and whose right leg is in $\mathcal{C}^\dagger$ (Barwick 14, section 5).)

## Examples

### Dress’ Mackey functors

For $\mathcal{A}$ taken to be (the derived category) of an abelian category (or better: postcomposed with a homological functor ) this definition reduces (Barwick 14) to that of Mackey functors as originally defined in (Dress 71).

### Equivariant spectra

Let $G$ be a finite group. Let $\mathcal{C}= G Set$ be its category of permutation representations. Then $Corr_1(\mathcal{C})$ is essentially what is called the Burnside category of $G$ (possibly after abelianizing/stabilizing the hom-spaces suitably, but as (Barwick 14) highlights, this is unnecessary when one is mapping out of this into something abelian/stable, as is the case here).

For $G$ finite then Mackey functors on $\mathcal{C}$ are equivalent to genuine G-spectra (Guillou-May 11, theorem 0.1, Barwick 14, below example B.6) (Notice that this equivalence does not in general hold if $G$ is not a finite group.)

(…)

For $E$ a genuine G-spectrum then the corresponding spectral Mackey functor is given by the equivariant homotopy groups of $E$

$G/H \mapsto E(G/H) = [\Sigma^\infty_+ G/H, E]_G \,,$

where on the right we have the $G$-equivariant mapping spectrum from the (equivariant) suspension spectrum of the orbit $G/H$ to $E$.

(e.g. (Guillou-May 11, remark 2.5), see also (Schwede 15, p. 16) for restriction and section 4 culminating on p. 37 for transfer and compatibility).

Further, the corresponding abelian-group valued Mackey functor is

$\pi_n(E) \colon G/H \mapsto [G/H_+\wedge S^n, X]_G \,,$

where now on the right we have just the homotopy classes of maps, i.e. the morphisms in the equivariant stable homotopy category (e.g. Greenlees-May 95, p. 43)

(…)

## Cohomology with coefficients in a Mackey functor

We discuss cohomology of topological G-spaces with coefficients in a Mackey functor, following notation and conventions as in (May 96, sections IX, X). See also (Greenlees-May 95, p. 9).

###### Defininition

For $X$ a pointed G-CW complex, define the chain complex $C_\bullet(X)$ of Mackey functors to be given by the stable equivariant homotopy groups of the quotient spaces $X^{\bullet}/X^{\bullet-1}$:

$C_n(X) \coloneqq \pi_n(X^n/X^{n-1}) \,,$

Then for $A$ any Mackey functor, the ordinary cohomology of $X$ with coefficients in $A$ is the cochain cohomology of the complex of homs of Mackey functors $C_n(X) \to A$:

$H_G^n(X,A) \coloneqq H^n( Hom(C_\bullet(X), A) ) \,.$

More generally, for $V$ a G-representation, the $(n-V)$-RO(G)-graded cohomology of $X$ with coefficients in $A$ is

$H_G^{n-V}(X,A) = H_G^n(S^V \wedge X, X) \,.$
###### Remark

The corresponding reduced cohomology $\tilde H^n(-,A)$ is represented by maps into the Eilenberg-MacLane G-space:

$\tilde H^n(X,A) \simeq [X,K(A,n)]_G \,.$
###### Remark

For this kind of cohomology, there is equivariant Serre spectral sequence (Kronholm 10).

## References

The original article is

• A. W. M. Dress, Notes on the theory of representations of finite groups. Part I: The Burnside ring of a finite group and some AGN-applications, Bielefeld, 1971,

Reviews and surveys include

• John Greenlees, Peter May, Equivariant stable homotopy theory, in I.M. James (ed.), Handbook of Algebraic Topology , pp. 279-325. 1995. (pdf)

• Peter May, section IX.4 of Equivariant homotopy and cohomology theory CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comeza˜na, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf)

• Peter Webb, A Guide to Mackey Functors (pdf)

• Michael Hill, Michael Hopkins, Douglas Ravenel, section 4 of The Arf-Kervaire problem in algebraic topology: Sketch of the proof (pdf)

(with an eye towards application to the Arf-Kervaire invariant problem)

• Megan Shulman, chapter 2 of Equivariant local coefficients and the $RO(G)$-graded cohomology of classifying spaces (arXiv:1405.1770)

• Tammo tom Dieck, Transformation groups, Studies in Mathematics, vol. 8, Walter de Gruyter, Berlin, New York, 1987, x + 311 pp.,

• Serge Bouc, chapter 1 of Green Functors and G-sets, LNM 1671 (1997; paperback 2008) doi:10.1007/BFb0095821

• Tammo tom Dieck, Equivariant homology and Mackey functors, Mathematische Annalen 206, no.1, pp. 67–78, 1973 doi:10.1007/BF01431529

• John Greenlees, Peter May, appendix A of Generalized Tate cohomology, Mem. Amer. Math. Soc. 113 (1995) no 543 (pdf)

• D. Tambara, The Drinfeld center of the category of Mackey functors, J. Algebra 319, 10, pp. 4018-4101 (2008) doi:10.1016/j.jalgebra.2008.02.011

• Elango Panchadcharam, Categories of Mackey Functors, PhD thesis, Macquarie Univ. 2006

• William Kronholm, The $RO(G)$-graded Serre spectral sequence, Homology Homotopy Appl. Volume 12, Number 1 (2010), 75-92. (pdf, Euclid)

The construction of equivariant stable homotopy theory in terms of Mackey functors is due to

Lectures notes include

Application of Mackey functors to the theory of motives includes

• Bruno Kahn, Takao Yamazaki, Voevodsky’s motives and Weil reciprocity, Duke Mathematical Journal 162, 14 (2013) 2751-2796 (arXiv:1108.2764)

Revised on August 13, 2017 10:49:40 by David Corfield (51.6.72.9)