The formal theory of relative monads
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915561685909504 |
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| author | Arkor, Nathanael McDermott, Dylan |
| author_facet | Arkor, Nathanael McDermott, Dylan |
| contents | We develop the theory of relative monads and relative adjunctions in a virtual equipment, extending the theory of monads and adjunctions in a 2-category. The theory of relative comonads and relative coadjunctions follows by duality. While some aspects of the theory behave analogously to the non-relative setting, others require new insights. In particular, the universal properties that define the algebra object and the opalgebra object for a monad in a virtual equipment are stronger than the classical notions of algebra object and opalgebra object for a monad in a 2-category. Inter alia, we prove a number of representation theorems for relative monads, establishing the unity of several concepts in the literature, including the devices of Walters, the $j$-monads of Diers, and the relative monads of Altenkirch, Chapman, and Uustalu. A motivating setting is the virtual equipment $\mathbb{V}\text{-}\mathbf{\mathbb{C}at}$ of categories enriched in a monoidal category $\mathbb{V}$, though many of our results are new even for $\mathbb{V} = \mathbf{Set}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_14014 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The formal theory of relative monads Arkor, Nathanael McDermott, Dylan Category Theory 18D70, 18D65, 18C15, 18C20, 18A40, 18D60, 18D20, 18N10, 18M65, 18M50 We develop the theory of relative monads and relative adjunctions in a virtual equipment, extending the theory of monads and adjunctions in a 2-category. The theory of relative comonads and relative coadjunctions follows by duality. While some aspects of the theory behave analogously to the non-relative setting, others require new insights. In particular, the universal properties that define the algebra object and the opalgebra object for a monad in a virtual equipment are stronger than the classical notions of algebra object and opalgebra object for a monad in a 2-category. Inter alia, we prove a number of representation theorems for relative monads, establishing the unity of several concepts in the literature, including the devices of Walters, the $j$-monads of Diers, and the relative monads of Altenkirch, Chapman, and Uustalu. A motivating setting is the virtual equipment $\mathbb{V}\text{-}\mathbf{\mathbb{C}at}$ of categories enriched in a monoidal category $\mathbb{V}$, though many of our results are new even for $\mathbb{V} = \mathbf{Set}$. |
| title | The formal theory of relative monads |
| topic | Category Theory 18D70, 18D65, 18C15, 18C20, 18A40, 18D60, 18D20, 18N10, 18M65, 18M50 |
| url | https://arxiv.org/abs/2302.14014 |