Effective homology for homotopy colimit and cofibrant replacement

TL;DR

This paper develops algorithms to compute homotopy colimits and cofibrant replacements of diagrams of simplicial sets with effective homology, enabling practical calculations in algebraic topology.

Contribution

It extends effective homology to diagrams of simplicial sets and provides algorithms for homotopy colimits and cofibrant replacements with effective homology.

Findings

Algorithms for homotopy colimit with effective homology

Algorithms for cofibrant replacement with effective homology

Application to equivariant cohomology operations

Abstract

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