Torsion Theories in a Non-pointed Context

TL;DR

This paper extends torsion theory concepts to non-pointed categories, exploring their connections with factorization systems and Galois structures, with diverse examples in algebra and topos theory.

Contribution

It introduces a non-pointed framework for torsion theories, linking them to factorization systems and Galois structures, with multiple illustrative examples.

Findings

Established a non-pointed torsion theory framework.

Connected torsion theories with factorization systems and Galois structures.

Provided examples in toposes, algebraic varieties, and abelian groups.

Abstract

We study a non-pointed version of the notion of torsion theory in the framework of categories equipped with a posetal monocoreflective subcategory such that the coreflector inverts monomorphisms. We explore the connections of such torsion theories with factorization systems and categorical Galois structures. We describe several examples of these torsion theories, in the dual of elementary toposes, in varieties of universal algebras used as models for non-classical logic, and in coslices of the category of abelian groups.