Fundamental group functors in descent-exact homological categories

TL;DR

This paper explores the concept of fundamental groups within descent-exact homological categories, providing generalized formulas for their computation across various algebraic and topological categories.

Contribution

It extends the notion of fundamental groups to a broad class of categories, including semi-abelian and topological algebras, with generalized Brown-Ellis-Hopf formulas.

Findings

Generalized Brown-Ellis-Hopf formulas for fundamental groups

Applicability to semi-abelian and topological categories

Unified framework for algebraic and topological fundamental groups

Abstract

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.