of abelian-group valued functors $\underline{M}=(M^*, M_*)$ on $\mcal{F}_{G}$, with $M^*$ contravariant and $M_*$ covariant. The functors $M^*$ and $M_*$ satisfy the following requirements.

1.  $M^{*}$,$M_{*}$ take the same value on objects and convert disjoint unions of $G$-sets into products of abelian groups.

2.  When $$\xymatrix{
    S \ar[r]^{\sigma'}\ar[d]^{\beta'} & T \ar[d]^{\beta} \\
    U \ar[r]^{\sigma}  & V 
    }$$ is a pull-back square of finite $G$-sets then $$\xymatrix{
    M(S) \ar[r]^{M_{*}(\sigma')} & M(T) \\
    M(U) \ar[u]^{M^{*}(\beta')} \ar[r]^{M_{*}(\sigma)} & M(V) \ar[u]^{M^{*}(\beta)}
    }$$ is a commutative square of abelian groups.

Given an abelian group $A$, the *constant Mackey functor* $\underline{A}$ is the Mackey functor which on objects $\underline{A}(G/K) = A$ and on a map $f:G/H\to G/K$, $M^*(f) = id$ and $M_*(f)$ is multiplication by the index $[K:H]$.

Bredon homology and cohomology
------------------------------

*Bredon cohomology (homology)* with coefficients in a Mackey functor $\underline{M}$ is a cohomology (homology) theory $H^{*}(-;\underline{M}))$ ($H_{*}(-;\underline{M})$ graded by $RO(G)$. For $V \in RO(G)$ there is an equivariant Eilenberg-Maclane space $K(\underline{M},V)$ which represents the reduced cohomology $\tilde{H}^{V}(X; \underline{M})$ for a based $G$-space, $$\tilde{H}^{V}(X; \underline{M}) = [X , K(\underline{M},V)]_{G} .$$

When $G=\Z/2$ then $RO(G) = \Z\oplus\Z$ with generators $\R^{1,0}$ and $\R^{0,1}$. We use the convention that $H^{p,q}(X;\underline{M}) = H^{\R^{p,q}}(X;\underline{M})$ (and similarly for homology).

If $A$ 