A relative monotone-light factorisation system for internal groupoids

TL;DR

This paper investigates the existence of a specific factorisation system for internal groupoids in exact categories, showing it exists under certain conditions related to Mal'tsev categories and Galois theory.

Contribution

It establishes that the comprehensive factorization acts as a relative monotone-light factorisation system in exact Mal'tsev categories, linking it to Galois theory and connected component reflectors.

Findings

No monotone-light factorisation system exists in general for internal groupoids.

In exact Mal'tsev categories, the comprehensive factorization provides the desired system.

Final functors and regular epimorphic discrete fibrations form the classes of the factorisation.

Abstract

Given an exact category $\mathcal{C}$, it is well known that the connected component reflector $\pi_0\colon\mathsf{Gpd}(\mathcal{C})\to\mathcal{C}$ from the category $\mathsf{Gpd}(\mathcal{C})$ of internal groupoids in $\mathcal{C}$ to the base category $\mathcal{C}$ is semi-left-exact. In this article we investigate the existence of a monotone-light factorisation system associated with this reflector. We show that, in general, there is no monotone-light factorisation system $(\mathcal{E}',\mathcal{M}^*)$ in $\mathsf{Gpd}(\mathcal{C})$, where $\mathcal{M}^*$ is the class of coverings in the sense of the corresponding Galois theory. However, when restricting to the case where $\mathcal{C}$ is an exact Mal'tsev category, we show that the so-called comprehensive factorization of regular epimorphisms in $\mathsf{Gpd}(\mathcal{C})$ is the relative monotone-light factorisation system (in the sense of Chikhladze) in the category $\mathsf{Gpd}(\mathcal{C})$ corresponding to the connected component reflector, where $\mathcal{E}'$ is the class of final functors and $\mathcal{M}^*$ the class of regular epimorphic discrete fibrations.