Ideally regular categories
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910036348895232 |
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| author | Mantovani, Sandra Messora, Mariano |
| author_facet | Mantovani, Sandra Messora, Mariano |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_21913 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ideally regular categories Mantovani, Sandra Messora, Mariano Category Theory 18E08, 18E13, 18C15, 08C15, 22A99 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. |
| title | Ideally regular categories |
| topic | Category Theory 18E08, 18E13, 18C15, 08C15, 22A99 |
| url | https://arxiv.org/abs/2511.21913 |