# Effective homology for homotopy colimit and cofibrant replacement

## Marek Filakovský

Address: Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, CZ 611 37 Brno, Czech Republic

E-mail: xfilakov@math.muni.cz

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

AMSclassification: primary 55N91; secondary 55U15.

Keywords: homotopy colimit, cofibrant replacement, effective homology, equivariant.

DOI: 10.5817/AM2014-5-273