Torsion theories and coverings of V-groups

TL;DR

This paper extends the theory of V-groups by establishing a torsion theory, characterizing coverings, and exploring their categorical properties, generalizing results from preordered groups to a broader algebraic setting.

Contribution

It introduces a torsion theory in V-groups, describes coverings as Galois groupoid actions, and connects these structures to pretorsion theories, broadening the understanding of V-group categories.

Findings

Existence of a torsion theory in V-Groups with specific subcategories

Characterization of coverings as Galois groupoid actions

Separated V-groups form a reflective and coreflective subcategory

Abstract

For a commutative, unital and integral quantale V, we generalize to V-groups the results developed by Gran and Michel for preordered groups. We first of all show that, in the category V-Grp of V-groups, there exists a torsion theory whose torsion and torsion-free subcategories are given by those of indiscrete and separated V-groups, respectively. It turns out that this torsion theory induces a monotone-light factorization system that we characterize, and it is then possible to describe the coverings in V-Grp. We next classify these coverings as internal actions of a Galois groupoid. Finally, we observe that the subcategory of separated V-groups is also a torsion-free subcategory for a pretorsion theory whose torsion subcategory is the one of symmetric V-groups. As recently proved by Clementino and Montoli, this latter category is actually not only coreflective, as it is the case for any torsion subcategory, but also reflective.