nLab Mackey functor

under construction

Idea

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. Much more relevant references are: this article of Weibel and Voevodsky’s seminal article defining his triangulated category of motives, specifically Section 3.4 about geometrical 0-motives).

Specifically, the concept was introduced and named as such in the context of representation theory (Dress 71) and equivariant homotopy theory/equivariant cohomology (May 96). Here the underlying category of correspondences is that in finite G-sets, called the Burnside category.

The equivariant homotopy groups $\pi_n(E)$ of a (genuine) G-spectrum $E$ organize into a Mackey functor on the Burnside category with values in abelian groups. This plays a key role in the equivariant Whitehead theorem. In fact, genunine G-spectra themselves are equivalent to Mackey (∞,1)-functors from the Burnside category to the (∞,1)-category of spectra (GuillouMay 11, Barwick 14).

Definition

In higher category theory

We follow the account in (Barwick 14), which incarnates a perspective of (Lindner 76) in higher category theory.

Let $\mathcal{C}$ be a disjunctive (∞,1)-category and write $\Span(\mathcal{C})$ for the (∞,1)-category of correspondences in $\mathcal{C}$ and let $\mathcal{A}$ be an (∞,1)-category with finite products.

Then an $\mathcal{A}$-valued semi-Mackey functor on $\mathcal{C}$ is a product-preserving functor

$$M \;\colon\; \Span(\mathcal{C}) \longrightarrow \mathcal{A} \,.$$

If $\mathcal{A}$ is additive, we say that $M$ is a Mackey functor.

Notice that this means that $M$ 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. satisfy the Beck-Chevalley condition.

In order to understand the Beck-Chevalley condition, note that disjunctiveness of $\mathcal{C}$ implies that $\Span(\mathcal{C})$ is semiadditive.

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

In representation theory

In order to recover the original definition of Dress 71, we fix $G$ a finite group, and we let $\mathbb{F}_G$ be the (disjunctive) category of finite $G$-sets. Then, the above definition specializes to $G$ by defining $G$-Mackey functors as product-preserving functors

$$M \;\colon\; \Span(\mathbb{F}_G) \longrightarrow \mathcal{A} \,.$$

$\Span(\mathbb{F}_G)$ has biproducts given by disjoint unions of finite $G$-sets; when $\mathcal{A}$ is a semiadditive category, such a functor may be decompiled into

1. a covariant functor $(-)_\ast \colon \mathcal{O}_G \to \mathcal{A}$ and

2. a contravariant functor $(-)^\ast \colon \mathcal{O}_G^{\op} \to \mathcal{A}$, subject to the conditions that

3. there exists a non-natural isomorphism $(-)_\ast \simeq (-)^\ast$, and

4. writing $R^H_J$ for the contravariant effect of in inclusion $J \subset H$ and $I^H_J$ for the covariant effect, $M$ satisfies the double coset formula

   $$R_J^H I_K^H(-) \simeq \bigoplus_{x \in [J\backslash H/K]} I_{J \cap ^x K}^J R_{J^x \cap K}^{K}(-)$$

   for all $J,K \subset H$, where $^x K \coloneqq xKx^{-1}$.

Examples

In representation theory

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). Fixing $G$ a finite group, we acquire the following examples (see section 3.2 of Blumberg 17).

1. Let $R \in \mathcal{A}$ be an object in $\mathcal{A}$. Then, the constant Mackey functor $\underline{R}$ assigns $R$ to every transitive $G$-set, whose restriction maps are all the identity, and whose transfer maps $G/H \rightarrow G/K$ are multiplication by $\left| H/K \right|$.

2. The Burnside Mackey functor $A(G)$ is defined by letting $A(G)([G/H])$ be the Grothendieck group of the cocartesian symmetric monoidal category of finite $H$-sets, with restriction and transfer given by restriction and induction, respectively.

3. The representation Mackey functor $R(G)$ is defined by letting $R(G)([G/H])$ be the Grothendieck group of the symmetric monoidal category of finite-dimensional $H$-representations, with restriction and transfer given by restriction and induction, respectively.

Additive completion of semi-Mackey functors

The free commutative monoid functor $\mathrm{Fr}\colon \mathcal{C} \rightarrow \mathrm{CMon}(\mathcal{C})$ is the unit of an adjunction whose left adjoint includes the category of (small) semiadditive categories? into categories with products; specifically, given $\mathcal{B}$ a semiadditive category, postcomposition with $\mathrm{Fr}$ induces an equivalence

$$\mathrm{Fun}^\times(\mathcal{B}, \mathcal{C}) \xrightarrow{\sim} \mathrm{Fun}^{\times} (\mathcal{B}, \mathrm{CMon}(\mathcal{C})),$$

and in particular, it induces an equivalence

$$\mathrm{sMack}(\mathcal{C}) \simeq \mathrm{sMack}(\mathrm{CMon}(\mathcal{C}).$$

We define Mackey functors in $\mathcal{C}$ to be the subcategory

$$\mathrm{Mack}(\mathcal{C}) \coloneqq \mathrm{sMack}(\mathrm{CGrp}(\mathcal{C})) \subset \mathrm{sMack}(\mathrm{CMon}(\mathcal{C})) \simeq \mathrm{sMack}(\mathcal{C})$$

We then have the following.

Proposition

Mackey functors form a reflective subcategory of semi-Mackey functors.

This follows formally from the fact that the group completion functor $(-)_{\simeq}\colon \mathrm{CGrp}(\mathcal{C}) \subset \mathrm{CMon}(\mathcal{C})$ is compatible with products; in particular, given $M$ a semi-Mackey functor, the value of $M_{\simeq}$ at the orbit $G/H$ is the group completion $M(G/H)_{\simeq}$, and the restrictions and transfers are gotten by group completing the restrictions and transfers of $M$.

Equivariant spectra

Let $G$ be a finite group. Let $\mathcal{C}= \mathbb{F}_G$ be the category of finite G-sets. 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, 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, the corresponding spectral Mackey functor is given by the fixed point spectra of $E$

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

where on the right we have the $G$-equivariant mapping spectrum from the equivariant suspension spectrum of the transitive G-set $G/H$ 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

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

(May 96, section X.4 def. 4.1, def. 4.2)

Remark

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

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

(Greenlees-May 95)

Remark

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

Related concepts

• Tambara functor

References

Plain Mackey functors

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,

The span perspective is due to

• Harald Lindner, A remark on Mackey-functors, 1976 (pdf)

Reviews and surveys:

• Andrew Blumberg, section 3.2 of The Burnside category, 2017 (pdf, GitHub)

• Tammo tom Dieck, Section II.9 of: Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)

• 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. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf)

• Stefan Schwede, around p. 16 of Lectures on Equivariant Stable Homotopy Theory, 2015 (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)

See also

• 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)

For Mackey functors enriched over closed multicategories:

• Niles Johnson, Donald Yau, Homotopy Theory of Enriched Mackey Functors (arXiv:2212.04276)

Relation of Mackey functors to sheaves with transfer in the theory of motives:

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

• Weibel, Charles. Transfer functors on k-algebras. Journal of Pure and Applied Algebra 201.1-3 (2005): 340-366

• Voevodsky, Vladimir. Triangulated categories of motives over a field. Cycles, transfers, and motivic homology theories 143 (2000): 188-238.

Categorification to Mackey 2-functors is discussed found in

• Paul Balmer, Ivo Dell'Ambrogio, Mackey 2-functors and Mackey 2-motives, (arXiv:1808.04902)

Spectral Mackey functors

The general concept of spectral Mackey functors

• Clark Barwick, Spectral Mackey functors and equivariant algebraic K-theory (I), Adv. Math., 304:646–727, 2017 (doi:10.1016/j.aim.2016.08.043, arXiv:1404.0108)

• Clark Barwick, Saul Glasman, Jay Shah, Spectral Mackey functors and equivariant algebraic K-theory (II) (arXiv:1505.03098)

• Saul Glasman, Stratified categories, geometric fixed points and a generalized Arone-Ching theorem (arXiv:1507.01976, talk notes pdf)

In equivariant stable homotopy theory

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

• Bert Guillou, Peter May, Models of $G$-spectra as presheaves of spectra, (arXiv:1110.3571)

• Bert Guillou, Peter May, Permutative $G$-categories in equivariant infinite loop space theory, Algebr. Geom. Topol. 17 (2017) 3259-3339 (arXiv:1207.3459)

• Denis Nardin, section 2.6 and appendix A of Stability and distributivity over orbital ∞-categories, 2012 (hdl.handle.net/1721.1/112895, pdf)

The generalization of K-theory of permutative categories to spectral Mackey functors is discussed in

• Anna Marie Bohmann, Angélica Osorno, Constructing equivariant spectra via categorical Mackey functors, Algebraic & Geometric Topology 15.1 (2015): 537-563 (arXiv:1405.6126)

In Goodwillie calculus

Discussion of Goodwillie calculus via spectral Mackey functors

• Saul Glasman, Goodwillie calculus and Mackey functors (arXiv:1610.03127)