Fundamental group functors in descent-exact homological categories

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