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.

Definition

Classical definition

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

For each subgroup , an abelian group ,
For each pair of subgroups with :
A restriction homomorphism ,
A transfer homomorphism .

These maps must satisfy the following axioms:

Functoriality: For nested subgroups , and .

Conjugation: For any and , there are isomorphisms compatible with restriction and transfer.

Double coset formula: For subgroups , the following identity holds:

:.

Modern definition

In modern category theory, a Mackey functor can be defined more elegantly using the language of spans. Let be a disjunctive quasi-category and be an additive quasi-category. A Mackey functor is a product-preserving functor where is the quasi-category of correspondences in .

Applications

In equivariant homotopy theory

Mackey functors play an important role in equivariant stable homotopy theory. For a genuine -spectrum , its equivariant homotopy groups form a Mackey functor given by:

where denotes morphisms in the equivariant stable homotopy category.

Cohomology with Mackey functor coefficients

For a pointed G-CW complex and a Mackey functor , one can define equivariant cohomology with coefficients in as:

where is the chain complex of Mackey functors given by stable equivariant homotopy groups of quotient spaces.