Mackey functor

Short description: Mathematical functor in representation theory and algebraic topology

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.

Definition

Classical definition

Let $G$ be a finite group. A Mackey functor $M$ for $G$ consists of:

• For each subgroup $H\le G$, an abelian group $M(H)$,
• For each pair of subgroups $H,K\le G$ with $H\subseteq K$:
  • A restriction homomorphism $R_H^K:M(K)\to M(H)$,
  • A transfer homomorphism $I_H^K:M(H)\to M(K)$.

These maps must satisfy the following axioms:

Functoriality: For nested subgroups $H\subseteq K\subseteq L$, $R_H^L=R_H^K\circ R_K^L$ and $I_H^L=I_K^L\circ I_H^K$.

Conjugation: For any $g\in G$ and $H\le G$, there are isomorphisms $c_g:M(H)\to M(gH{g}^{-1})$ compatible with restriction and transfer.

Double coset formula: For subgroups $H,K\le G$, the following identity holds:

$R_H^G{I}_K^G=\sum _{x\in [H\mathrm{\setminus }G/K]}{I}_{H\cap xK{x}^{-1}}^H\circ c_x\circ R_{x^{-1}Hx\cap K}^K$.

Modern definition

In modern category theory, a Mackey functor can be defined more elegantly using the language of spans. Let ${𝒞}$ be a disjunctive quasi-category and ${𝒜}$ be an additive quasi-category. A Mackey functor is a product-preserving functor $M:\text{Span}({𝒞})\to {𝒜}$ where $\text{Span}({𝒞})$ is the quasi-category of correspondences in ${𝒞}$.

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:

${\pi }_n(E):G/H\mapsto [G/H_{+}\wedge S^n,X{]}^G$

where $[-,-{]}^G$ denotes morphisms in the equivariant stable homotopy category.

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:

$H_G^n(X,A):=H^n(\text{Hom}(C_{\bullet }(X),A))$

where $C_{\bullet }(X)$ is the chain complex of Mackey functors given by stable equivariant homotopy groups of quotient spaces.

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.