Mackey Functor

In mathematics, particularly in representation theory and algebraic topology, a Mackey functor is a type of functor that generalizes various constructions in group theory and equivariant homotopy theory. Named after American mathematician George Mackey, these functors were first introduced by German mathematician Andreas Dress in 1971.Dress, A. W. M. (1971). "Notes on the theory of representations of finite groups. Part I: The Burnside ring of a finite group and some AGN-applications". Bielefeld.

Definition

Classical definition

Let $G$ be a finite group. A Mackey functor $M$ for $G$ consists of:
* For each subgroup $H \leq G$, an abelian group $M(H)$,
* For each pair of subgroups $H, K \leq G$ with $H \subseteq K$:
** A restriction homomorphism $R^K_H: M(K) \to M(H)$,
** A transfer homomorphism $I^K_H: M(H) \to M(K)$.
These maps must satisfy the following axioms:
:Functoriality: For nested subgroups $H \subseteq K \subseteq L$, $R^L_H = R^K_H \circ R^L_K$ and $I^L_H = I^L_K \circ I^K_H$.
:Conjugation: For any $g \in G$ and $H \leq G$, there are isomorphisms $c_g: M(H) \to M(gHg^)$ compatible with restriction and transfer.
:Double coset formula: For subgroups $H, K \leq G$, the following identity holds:
::$R^G_H I^G_K = \sum_ I^H_ \circ c_x \circ R^K_$.

Modern definition

In modern category theory, a Mackey functor can be defined more elegantly using the language of spans. Let $\mathcal$ be a disjunctive $(\infty, 1)$-category and $\mathcal$ be an additive $(\infty, 1)$-category ($(\infty, 1)$-categories are also known as quasi-categories). A Mackey functor is a product-preserving functor $M: \text(\mathcal) \to \mathcal$ where $\text(\mathcal)$ is the $(\infty, 1)$-category of correspondences in $\mathcal$.Barwick, C. (2017). "Spectral Mackey functors and equivariant algebraic K-theory (I)". ''Advances in Mathematics'', 304:646–727.

Applications

In equivariant homotopy theory

Mackey functors play an important role in equivariant stable homotopy theory. For a genuine $G$-spectrum $E$, its equivariant homotopy groups form a Mackey functor given by:
:\pi_n(E): G/H \mapsto /H_+ \wedge S^n, XG
where ,-G denotes morphisms in the equivariant stable homotopy category.May, J. P. (1996). "Equivariant homotopy and cohomology theory". ''CBMS Regional Conference Series in Mathematics'', vol. 91.

Cohomology with Mackey functor coefficients

For a pointed G-CW complex $X$ and a Mackey functor $A$, one can define equivariant cohomology with coefficients in $A$ as:
:$H^n_G(X,A) := H^n(\text(C_\bullet(X), A))$
where $C_\bullet(X)$ is the chain complex of Mackey functors given by stable equivariant homotopy groups of quotient spaces.Kronholm, W. (2010). "The RO(G)-graded Serre spectral sequence". ''Homology, Homotopy and Applications'', 12(1):75-92.

References

Further reading

*Dieck, T. (1987). ''Transformation Groups''. de Gruyter. {{ISBN, 978-3110858372
*Webb, P. "A Guide to Mackey Functors"
*Bouc, S. (1997). "Green Functors and G-sets". ''Lecture Notes in Mathematics'' 1671. Springer-Verlag.

Representation theory
Algebraic topology
Functors
Homological algebra