Effective homology for homotopy colimit and cofibrant replacement

Abstract

We extend the notion of simplicial set with effective homology presented in [22] to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X \colon \mathcal{I}\rightarrow \mbox{sSet}$ such that each simplicial set $X(i)$ has effective homology, we present an algorithm computing the homotopy colimit $\mbox{hocolim}\,X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement $X^{\mbox{cof}}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology operations.