A description of the fundamental group in terms of commutators and closure operators

TL;DR

This paper links Galois theory and closure operators to develop a generalized Hopf formula, enabling new calculations of fundamental groups across various algebraic categories.

Contribution

It introduces a novel connection between semi-abelian homology, Galois theory, and closure operators, broadening the tools for computing fundamental groups.

Findings

Derived a generalized Hopf formula for homology.

Enabled calculation of fundamental groups in multiple algebraic categories.

Linked Galois-theoretic approach with homological closure operators.

Abstract

A connection between the Galois-theoretic approach to semi-abelian homology and the homological closure operators is established. In particular, a generalised Hopf formula for homology is obtained, allowing the choice of a new kind of functors as coefficients. This makes it possible to calculate the fundamental groups corresponding to many interesting reflections arising, for instance, in the categories of groups, rings, compact groups and simplicial loops.