Torsion theories and coverings of preordered groups

TL;DR

This paper investigates a non-abelian torsion theory in preordered groups, analyzing its structure, coverings, and connections to Galois theory, and introduces a pretorsion theory related to internal groups.

Contribution

It introduces a new non-abelian torsion theory for preordered groups, describes its Galois coverings, and connects it to internal group structures and recent categorical discoveries.

Findings

The reflector has stable units and induces a monotone-light factorization system.

Coverings are classified as internal actions of a Galois groupoid.

A pretorsion theory identifies the subcategory of internal groups as protomodular objects.

Abstract

In this article we explore a non-abelian torsion theory in the category of preordered groups: the objects of its torsion-free subcategory are the partially ordered groups, whereas the objects of the torsion subcategory are groups (with the total order). The reflector from the category of preordered groups to this torsion-free subcategory has stable units, and we prove that it induces a monotone-light factorization system. We describe the coverings relative to the Galois structure naturally associated with this reflector, and explain how these coverings can be classified as internal actions of a Galois groupoid. Finally, we prove that in the category of preordered groups there is also a pretorsion theory, whose torsion subcategory can be identified with a category of internal groups. This latter is precisely the subcategory of protomodular objects in the category of preordered groups, as recently discovered by Clementino, Martins-Ferreira, and Montoli.