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

Abstract

This paper considers non-Abelian homology groups of a group diagram introduced as homotopy groups of a simplicial change. We prove a theorem stating that the non-Abelian homology groups of a group diagram are isomorphic to the homotopy groups of the homotopy colimit of a classifying space diagram, with the dimension shifted by 1. Bousfield and Kan proved an isomorphism between the homotopy groups of an Abelian simplicial group and the homology groups of this simplicial group. We generalize this to non-Abelian simplicial groups. We also develop a method for finding a non-zero homotopy group of smallest dimension for the homotopy colimit of classifying spaces. For a group diagram over a free category with a zero colimit, we obtain a criterion for the isomorphism of the first non-Abelian and Abelian homology groups.