Homotopy cofinality for Non-Abelian homology of group diagrams

TL;DR

This paper develops a homotopy cofinality framework for non-Abelian homology of group diagrams, establishing invariance properties and constructing a generalized homology theory for presheaves over small categories.

Contribution

It introduces a homotopy cofinality approach to non-Abelian homology, extending it to presheaves and establishing isomorphisms with Baues-Wirsching homology.

Findings

Homotopy cofinal functors induce weak equivalences between homotopy colimits.

Non-Abelian homology is invariant under left Kan extension along virtual discrete cofibrations.

A generalized non-Abelian Gabriel-Zisman homology theory for simplicial sets and presheaves is constructed.

Abstract

We prove that a homotopy cofinal functor between small categories induces a weak equivalence between homotopy colimits of pointed simplicial sets. This is used to prove that the non-Abelian homology of a group diagram is isomorphic to the homology of its inverse image under a homotopy cofinal functor. This also made it possible to establish that the non-Abelian homology of group diagrams are invariant under the left Kan extension along virtual discrete cofibrations. With the help of these results, we have constructed a non-Abelian Gabriel-Zisman homology theory for simplicial sets with coefficients in group diagrams. Moreover, we have generalized this theory to presheaves over an arbitrary small category, which play the role of simplicial sets. Sufficient conditions are found for the isomorphism of non-Abelian homology of presheaves over a small category with coefficients in group diagrams. It is proved that the non-Abelian homology of the factorization category for the small category is isomorphic to the Baues-Wirsching homology. A condition is found under which a functor between small categories induces an isomorphism of non-Abelian Baues-Wirsching homology.