Non-Abelian homology and homotopy colimit of classifying spaces for a diagram of groups

TL;DR

The paper develops non-Abelian homology for diagrams of groups by using a simplicial replacement and proves a fundamental isomorphism ${\rm colim}^{\mathscr C}_n {\cal G} \cong \pi_{n+1}({\rm hocolim}^{\mathscr C} B{\cal G})$, linking diagram homology to homotopy groups of a homotopy colimit. It shows these non-Abelian homology groups coincide with cotriple derived functors of the colimit functor, using a cotriple $T=\Lambda\Omega$ to construct resolutions and derive $L_n {\rm colim}^{\mathscr C}$. The work provides a computable framework: a criterion for when the first non-Abelian and Abelian homology groups agree for diagrams over a free category with zero colimit, and a method to extract least-dimension nonzero homotopy groups of hocolims of classifying spaces. Overall, the results unify non-Abelian diagram homology with homotopy-theoretic constructions and cotriple homology, offering practical tools for calculating homotopy groups of hocolims in algebraic topology.

Abstract

The paper considers non-Abelian homology groups for a diagram of groups introduced as homotopy groups of a simplicial replacement. It is proved that the non-Abelian homology groups of the group diagram are isomorphic to the homotopy groups of the homotopy colimit of the diagram of classifying spaces, with a dimension shift of 1. As an application, a method is developed for finding a nonzero homotopy group of least dimension for a homotopy colimit of classifying spaces. For a group diagram over a free category with a zero colimit, a criterion for the isomorphism of the first non-Abelian and first Abelian homology groups is obtained. It is established that the non-Abelian homology groups are isomorphic to the cotriple derived functors of the colimit functor defined on the category of group diagrams.