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.
Keywords: Ideally regular category, ideally exact category, ideal, homological category, quasivariety, topological variety
2020 MSC: 18E08, 18E13, 18C15, 08C15, 22A99
Theory and Applications of Categories, Vol. 45, 2026, No. 11, pp 391-400.
Published 2026-02-24.
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