Bredon cohomology

Definition

Let $G$ be a discrete group. The orbit category $\mathcal{O}_G$ has as objects the left $G$-sets $G/H$ for subgroups $H \le G$, and as morphisms the $G$-equivariant maps. A coefficient system is a contravariant functor $M \colon \mathcal{O}_G \to \mathrm{Ab}$ to abelian groups.

Let $X$ be a $G$-CW complex, that is, a CW complex with a cellular action of $G$. It is required that each element of $G$ that maps a cell to itself fixes that cell pointwise. For each subgroup $H$, the fixed set $X^H$ is a subcomplex. For $n \ge 0$, let $\underline{C}_n(X)$ be the contravariant functor $\mathcal{O}_G \to \mathrm{Ab}$ given by $G/H \mapsto C_n(X^H)$, the cellular chains of $X^H$. A $G$-map $G/H \to G/K$ with $H \mapsto gK$ acts by $C_n(X^K) \to C_n(X^H)$, $c \mapsto g c$. The cellular boundary maps make $\underline{C}_\bullet(X)$ a chain complex of functors. The Bredon cochain complex is $$C^n_G(X; M) = \operatorname{Nat}(\underline{C}_n(X), M),$$ the group of natural transformations, with coboundary induced by the boundary of $\underline{C}_\bullet(X)$. The Bredon cohomology $H^n_G(X; M)$ is the $n$-th cohomology of $C^\bullet_G(X; M)$.