Semi-localizations of semi-abelian categories

TL;DR

This paper characterizes semi-localizations within semi-abelian categories, linking them to torsion theories and providing new insights into their structure through categorical and relational methods.

Contribution

It offers a novel characterization of semi-localizations in semi-abelian categories and introduces a new binary relation-based characterization of protomodular categories.

Findings

Characterization of semi-localizations of exact Mal'tsev categories

Identification of torsion-free subcategories in semi-abelian categories

Examples in groups, rings, and topological groups

Abstract

A semi-localization of a category is a full reflective subcategory with the property that the reflector is semi-left-exact. In this article we first determine an abstract characterization of the categories which are semi-localizations of an exact Mal'tsev category, by specializing a result due to S. Mantovani. We then turn our attention to semi-abelian categories, where a special type of semi-localizations are known to coincide with torsion-free subcategories. A new characterisation of protomodular categories in terms of binary relations is obtained, inspired by the one discovered in the pointed context by Z. Janelidze. This result is useful to obtain an abstract characterization of the torsion-free and of the hereditarily-torsion-free subcategories of semi-abelian categories. Some examples are considered in detail in the categories of groups, crossed modules, commutative rings and topological groups. We finally explain how these results extend similar ones obtained by W. Rump in the abelian context.