  We study the notion of fundamental group in the framework of descent-exact
homological categories. This setting is sufficiently wide to include several
categories of "algebraic" nature such as the almost abelian categories, the
semi-abelian categories, and the categories of topological semi-abelian
algebras. For many adjunctions in this context, the fundamental groups are
described by generalised Brown-Ellis-Hopf formulae for the integral homology of
groups.
