Profunctorial algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914463799574528 |
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| author | Aristote, Quentin Tarantino, Umberto |
| author_facet | Aristote, Quentin Tarantino, Umberto |
| contents | We provide a bicategorical generalization of Barr's landmark 1970 paper, in which he describes how to extend Set-monads to relations and uses this to characterize topological spaces as the relational algebras of the ultrafilter monad. With two-sided discrete fibrations playing the role of relations in a bicategory, we first describe how to extend pseudomonads on a bicategory to skew monads on its bicategory of two-sided discrete fibrations, and we characterize in terms of exact squares when these extensions are themselves pseudomonads. As a wide class of examples, we show that every Set-monad induces a pseudomonad on the 2-category of categories admitting a skew extension to profunctors, and in a few relevant cases we introduce suitable quotients also extending to profunctors. Among the latter, we then focus on the ultracompletion pseudomonad, whose pseudoalgebras are ultracategories: we characterize the normalized lax algebras of its profunctorial extension as ultraconvergence spaces, a recently-introduced categorification of topological spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22721 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Profunctorial algebras Aristote, Quentin Tarantino, Umberto Category Theory Logic in Computer Science 18N15 (Primary) 18C10, 03G30, 03C20, 54D80 (Secondary) We provide a bicategorical generalization of Barr's landmark 1970 paper, in which he describes how to extend Set-monads to relations and uses this to characterize topological spaces as the relational algebras of the ultrafilter monad. With two-sided discrete fibrations playing the role of relations in a bicategory, we first describe how to extend pseudomonads on a bicategory to skew monads on its bicategory of two-sided discrete fibrations, and we characterize in terms of exact squares when these extensions are themselves pseudomonads. As a wide class of examples, we show that every Set-monad induces a pseudomonad on the 2-category of categories admitting a skew extension to profunctors, and in a few relevant cases we introduce suitable quotients also extending to profunctors. Among the latter, we then focus on the ultracompletion pseudomonad, whose pseudoalgebras are ultracategories: we characterize the normalized lax algebras of its profunctorial extension as ultraconvergence spaces, a recently-introduced categorification of topological spaces. |
| title | Profunctorial algebras |
| topic | Category Theory Logic in Computer Science 18N15 (Primary) 18C10, 03G30, 03C20, 54D80 (Secondary) |
| url | https://arxiv.org/abs/2601.22721 |