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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 $X:ℐ\to \text{sSet}$ such that each simplicial set $X\left(i\right)$ has effective homology, we present an algorithm computing the homotopy colimit $\text{hocolim}X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement $X^{\text{cof}}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology operations....

In the paper weak sufficient conditions for the reduction of the chain complex of a twisted cartesian product $F{\times }_{\tau }B$ to a chain complex of free finitely generated abelian groups are found.