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.^[1][2]

Definition

Classical definition

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

• For each subgroup ${\displaystyle H\leq G}$, an abelian group ${\displaystyle M(H)}$,
• For each pair of subgroups ${\displaystyle H,K\leq G}$ with ${\displaystyle H\subseteq K}$:
  • A restriction homomorphism ${\displaystyle R_{H}^{K}:M(K)\to M(H)}$,
  • A transfer homomorphism ${\displaystyle I_{H}^{K}:M(H)\to M(K)}$.

These maps must satisfy the following axioms:

Functoriality: For nested subgroups ${\displaystyle H\subseteq K\subseteq L}$, ${\displaystyle R_{H}^{L}=R_{H}^{K}\circ R_{K}^{L}}$ and ${\displaystyle I_{H}^{L}=I_{K}^{L}\circ I_{H}^{K}}$.

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

Double coset formula: For subgroups ${\displaystyle H,K\leq G}$, the following identity holds:

${\displaystyle R_{H}^{G}I_{K}^{G}=\sum _{x\in [H\backslash G/K]}I_{H\cap xKx^{-1}}^{H}\circ c_{x}\circ R_{x^{-1}Hx\cap K}^{K}}$.^[1]

Modern definition

In modern category theory, a Mackey functor can be defined more elegantly using the language of spans. Let ${\displaystyle {\mathcal {C}}}$ be a disjunctive ${\displaystyle (\infty ,1)}$-category and ${\displaystyle {\mathcal {A}}}$ be an additive ${\displaystyle (\infty ,1)}$-category (${\displaystyle (\infty ,1)}$-categories are also known as quasi-categories). A Mackey functor is a product-preserving functor ${\displaystyle M:{\text{Span}}({\mathcal {C}})\to {\mathcal {A}}}$ where ${\displaystyle {\text{Span}}({\mathcal {C}})}$ is the ${\displaystyle (\infty ,1)}$-category of correspondences in ${\displaystyle {\mathcal {C}}}$.^[3]

Applications

In equivariant homotopy theory

Mackey functors play an important role in equivariant stable homotopy theory. For a genuine ${\displaystyle G}$-spectrum ${\displaystyle E}$, its equivariant homotopy groups form a Mackey functor given by:

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

where ${\displaystyle [-,-]^{G}}$ denotes morphisms in the equivariant stable homotopy category.^[4]

Cohomology with Mackey functor coefficients

For a pointed G-CW complex ${\displaystyle X}$ and a Mackey functor ${\displaystyle A}$, one can define equivariant cohomology with coefficients in ${\displaystyle A}$ as:

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

where ${\displaystyle C_{\bullet }(X)}$ is the chain complex of Mackey functors given by stable equivariant homotopy groups of quotient spaces.^[5]