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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.06420 |
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| _version_ | 1866909319409172480 |
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| author | Hirata, Kengo |
| author_facet | Hirata, Kengo |
| contents | We study lax functors between bicategories as a generalized concept of monads and describe generalized notions and theorems of formal monad theory for lax functors. Our first approach is to use the 2-monad whose lax algebras are lax functors. We define lax doctrinal adjunctions for a 2-monad $T$ on a 2-category $\mathcal{K}$, and we show that if $\mathcal{K}$ admits and $T$ preserves certain codescent objects, the 2-category $\mathrm{Lax}\text{-}{T}\text{-}\mathrm{Alg}_{c}$ of lax algebras and colax morphisms can coreflectively be embedded in the 2-category of lax doctrinal adjunctions. This coreflective embedding generalizes the relation between monads and adjunctions. Our second approach is to see a distributive law for monads as a 2-functor from a lax Gray tensor product, and we show a generalized form of Beck's characterization of distributive laws. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_06420 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generalization of formal monad theory to lax functors Hirata, Kengo Category Theory 18N10, 18N15 We study lax functors between bicategories as a generalized concept of monads and describe generalized notions and theorems of formal monad theory for lax functors. Our first approach is to use the 2-monad whose lax algebras are lax functors. We define lax doctrinal adjunctions for a 2-monad $T$ on a 2-category $\mathcal{K}$, and we show that if $\mathcal{K}$ admits and $T$ preserves certain codescent objects, the 2-category $\mathrm{Lax}\text{-}{T}\text{-}\mathrm{Alg}_{c}$ of lax algebras and colax morphisms can coreflectively be embedded in the 2-category of lax doctrinal adjunctions. This coreflective embedding generalizes the relation between monads and adjunctions. Our second approach is to see a distributive law for monads as a 2-functor from a lax Gray tensor product, and we show a generalized form of Beck's characterization of distributive laws. |
| title | Generalization of formal monad theory to lax functors |
| topic | Category Theory 18N10, 18N15 |
| url | https://arxiv.org/abs/2301.06420 |