known as *contractible slice cover*, is guaranteed for proper $G$-ANR’s by an appropriate version of the Slice Theorem (see [@antonyan]).

Let ${\mathcal{N}_{G} \mathcal{U}}$ be the category where the objects are non-empty finite intersections of $\mathcal{U}$, and where a morphism is an inclusion.

A covariant system with values on $R$-Modules is a covariant functor ${\mathcal{N}_{G} \mathcal{U}}\to  R-{\rm Mod}$.

The nerve associated to the cover $\mathcal{U}$, denoted by ${\mathcal{N}_{G} \mathcal{U}}^*$ is the simplicial set where the set of $0$-simplices ${\mathcal{N}_{G} \mathcal{U}}^0$ is given by elements of the cover $\mathcal{U}$. The $1$ -simplices in ${\mathcal{N}_{G} \mathcal{U}}^{1}$ are non-empty intersections $U_{\sigma_{1}}\cap U_{\sigma_{2}}$ an in general, for any natural number $n$, the $n$-simplices, ${\mathcal{N}_{G} \mathcal{U}}^{n}$ are non-empty intersections of elements in the cover $\mathcal{U}$ of length $n+1$, $U_{\sigma_1}\cap \ldots \cap U_{\sigma_2}$.

There are face $\delta_i: {\mathcal{N}_{G} \mathcal{U}}^{i}\to {\mathcal{N}_{G} \mathcal{U}}^{i-1}$ and degeneracy maps $s_i: {\mathcal{N}_{G} \mathcal{U}}^{i-1}\to {\mathcal{N}_{G} \mathcal{U}}^i$ coming from the associativity, respectively idempotence of intersection.

Let $X$ be a proper $G$-space satisfying condition \[conditionCMP\]. Given a contractible slice cover $\mathcal{U}$, and a covariant coefficient system $M$, the Bredon equivariant homology groups with respect to $\mathcal{U}$, denoted by $H_{n}^G(X,\mathcal{U}; M),$ are the homology groups of the chain complex defined as the $R$-module $$C_*( {\mathcal{N}_{G} \mathcal{U}}, M)$$

