Generalized Eilenberg-Zilber type theorem and its equivariant applications (Q1299972)

If \(\mathcal C\) is a small category and \(F\) is a functor from \({\mathcal C}\) to the category of \(R\)-modules for some ring \(R\), then the the homology \(H_\ast (\mathcal C,F)\) of \(\mathcal C\) with coefficients in \(F\) is defined here as the cohomology of the nerve (or classifying space) of \(\mathcal C\) with coefficients in the local coefficient system generated from \(F\). Here \(R\)-modules can be left \(R\)-modules, right \(R\)-modules, or bimodules, as you will. As such there is an Eilenberg-Zilber Theorem for computing \(H_\ast(\mathcal C \times {\mathcal C}',F \otimes_R F')\), whenever the tensor product makes sense. Because the nerve of the product of two small categories is the product of the nerves, any of the standard proofs would work, and the authors outline one of them here. An application is given to Bredon homology with local coefficients of a \(G\) space \(X\), where \(G\) is a discrete group. Indeed, Bredon homology can be realized as the homology of the equivariant simplex category of \(X\) with coefficients in some \(F\) as above.