  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.
