Ideally regular categories

TL;DR

This paper introduces ideally regular categories, a generalization of ideally exact categories, expanding the framework to include new algebraic and topological structures with applications in universal algebra and topology.

Contribution

It defines ideally regular categories, characterizes them via monadicity over homological categories, and provides diverse examples including algebraic and topological varieties.

Findings

Includes categories of torsion-free unital rings and topological rings.

Shows all semi-localisations of ideally exact categories are ideally regular.

Provides a unifying framework for algebraic and topological categories.

Abstract

In this note, we propose a generalisation of G. Janelidze's notion of an ideally exact category beyond the Barr exact setting. We define an ideally regular category as a regular, Bourn protomodular category with finite coproducts in which the unique morphism 0 -> 1 is effective for descent. As in the ideally exact case, ideally regular categories support a notion of ideal that classifies regular quotients. Moreover, they admit a characterisation in terms of monadicity over a homological category (rather than a semi-abelian one, as in the exact setting). Examples include Bourn protomodular quasivarieties of universal algebra in which 0 -> 1 is effective for descent (such as the category of torsion-free unital rings), all Bourn protomodular topological varieties with at least one constant (such as topological rings), and all semi-localisations of ideally exact categories.