Ideally regular categories
Sandra Mantovani, Mariano Messora
TL;DR
The paper generalises Janelidze's ideally exact categories to the broader notion of ideally regular categories. It defines ideals in this setting and proves that they classify regular quotients, mirroring the ideal-exact theory but without Barr exactness. A central contribution is a monadic characterisation over a homological category, replacing the semi-abelian requirement in the exact case. The authors present concrete instances, including torsion-free unital rings, topological unital rings, and semi-localisations, and they highlight the essential role of the effective descent condition through a counterexample.
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.