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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).
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).
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
Notice that this means that $S$ is in particular
a covariant (∞,1)-functor $(-)_\ast \colon\mathcal{C} \to \mathcal{A}$;
a contravariant (∞,1)-functor, hence $(-)^\ast \colon\mathcal{C}^{op} \to \mathcal{A}$;
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).)
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).
Let $G$ be a finite group. Let $\mathcal{C}= G Set$ be its category of 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 $E$
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
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)
(…)
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).
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 $X^{\bullet}/X^{\bullet-1}$:
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 $C_n(X) \to A$:
More generally, for $V$ a G-representation, the $(n-V)$-RO(G)-graded cohomology of $X$ with coefficients in $A$ is
(May 96, section X.4 def. 4.1, def. 4.2)
The corresponding reduced cohomology $\tilde H^n(-,A)$ is represented by maps into the Eilenberg-MacLane G-space:
For this kind of cohomology, there is equivariant Serre spectral sequence (Kronholm 10).
The original article is
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)
Relation of Mackey functors to sheaves with transfer in the theory of motives:
Categorification to Mackey 2-functors is discussed found in
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)
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
Discussion of Goodwillie calculus via spectral Mackey functors
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