Quite generally, the term Mackey functor refers to an additive functor from a (subcategory of) a category of correspondences (in a disjunctive category ) 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 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). In that context it first received its full (∞,1)-category-theoretic formulation and discussion (Barwick 14).
We follow the modern account in (Barwick 14).
Let be a disjunctive (∞,1)-category and write for the (∞,1)-category of correspondences in , regarded as a symmetric monoidal (∞,1)-category with respect to its coproduct (which is a biproduct by disjunctiveness of ).
Write Spectra for the (∞,1)-category of spectra regarded as a symmetric monoidal (∞,1)-category with respect to the smash product of spectra. More generally could be any symmetric monoidal stable (∞,1)-category
Then a (spectral) Mackey functor on is a monoidal (∞,1)-functor of the form
Notice that this means that is in particular
a covariant (∞,1)-functor ;
a contravariant (∞,1)-functor, hence ;
satisfying the Beck-Chevalley condition.
For 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 be a finite group. Let be its category of permutation representations. Then is essentially what is called the Burnside category of (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 finite then Mackey functors on 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 is not a finite group.)
(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 compatibility5))
The original article is
Reviews and surveys include
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)
(with an eye towards application to the Arf-Kervaire invariant problem)
Tammo tom Dieck, Transformation groups, Studies in Mathematics, vol. 8, Walter de Gruyter, Berlin, New York, 1987, x + 311 pp.,
chapter 1 of Serge Bouc, Green Functors and G-sets, LNM 1671 (1997; paperback 2008) doi:10.1007/BFb0095821
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
Permutative -categories in equivariant infinite loop space theory (arXiv:1207.3459)
Lectures notes include
Application of Mackey functors to the theory of motives includes