Bicategories of algebras for relative pseudomonads
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909463216128000 |
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| author | Arkor, Nathanael Saville, Philip Slattery, Andrew |
| author_facet | Arkor, Nathanael Saville, Philip Slattery, Andrew |
| contents | We introduce pseudoalgebras for relative pseudomonads and develop their theory. For each relative pseudomonad $T$, we construct a free--forgetful relative pseudoadjunction that exhibits the bicategory of $T$-pseudoalgebras as terminal among resolutions of $T$. The Kleisli bicategory for $T$ thus embeds into the bicategory of pseudoalgebras as the sub-bicategory of free pseudoalgebras. We consequently obtain a coherence theorem that implies, for instance, that the bicategory of distributors is biequivalent to the 2-category of presheaf categories. In doing so, we extend several aspects of the theory of pseudomonads to relative pseudomonads, including doctrinal adjunction, transport of structure, and lax-idempotence. As an application of our general theory, we prove that, for each class of colimits $Φ$, there is a correspondence between monads relative to free $Φ$-cocompletions, and $Φ$-cocontinuous monads on free $Φ$-cocompletions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12510 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bicategories of algebras for relative pseudomonads Arkor, Nathanael Saville, Philip Slattery, Andrew Category Theory 18C15, 18C20, 18D60, 18M50, 18N10, 18N15, 18N20 We introduce pseudoalgebras for relative pseudomonads and develop their theory. For each relative pseudomonad $T$, we construct a free--forgetful relative pseudoadjunction that exhibits the bicategory of $T$-pseudoalgebras as terminal among resolutions of $T$. The Kleisli bicategory for $T$ thus embeds into the bicategory of pseudoalgebras as the sub-bicategory of free pseudoalgebras. We consequently obtain a coherence theorem that implies, for instance, that the bicategory of distributors is biequivalent to the 2-category of presheaf categories. In doing so, we extend several aspects of the theory of pseudomonads to relative pseudomonads, including doctrinal adjunction, transport of structure, and lax-idempotence. As an application of our general theory, we prove that, for each class of colimits $Φ$, there is a correspondence between monads relative to free $Φ$-cocompletions, and $Φ$-cocontinuous monads on free $Φ$-cocompletions. |
| title | Bicategories of algebras for relative pseudomonads |
| topic | Category Theory 18C15, 18C20, 18D60, 18M50, 18N10, 18N15, 18N20 |
| url | https://arxiv.org/abs/2501.12510 |