What is the universal property of the 2-category of monads?
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913412278124544 |
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| author | Lack, Stephen Miranda, Adrian |
| author_facet | Lack, Stephen Miranda, Adrian |
| contents | For a 2-category $\mathcal{K}$, we consider Street's 2-category Mnd($\mathcal{K}$) of monads in $\mathcal{K}$, along with Lack and Street's 2-category EM($\mathcal{K}$) and the identity-on-objects-and-1-cells 2-functor Mnd($\mathcal{K}$) $\to$ EM($\mathcal{K}$) between them. We show that this 2-functor can be obtained as a "free completion" of the 2-functor $1\colon \mathcal{K} \to \mathcal{K}$. We do this by regarding 2-functors which act as the identity on both objects and 1-cells as categories enriched a cartesian closed category $\mathbf{BO}$ whose objects are identity-on-objects functors. We also develop some of the theory of $\mathbf{BO}$-enriched categories. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_02210 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | What is the universal property of the 2-category of monads? Lack, Stephen Miranda, Adrian Category Theory 18C15, 18C20, 18D20, 18N10, 18A35 For a 2-category $\mathcal{K}$, we consider Street's 2-category Mnd($\mathcal{K}$) of monads in $\mathcal{K}$, along with Lack and Street's 2-category EM($\mathcal{K}$) and the identity-on-objects-and-1-cells 2-functor Mnd($\mathcal{K}$) $\to$ EM($\mathcal{K}$) between them. We show that this 2-functor can be obtained as a "free completion" of the 2-functor $1\colon \mathcal{K} \to \mathcal{K}$. We do this by regarding 2-functors which act as the identity on both objects and 1-cells as categories enriched a cartesian closed category $\mathbf{BO}$ whose objects are identity-on-objects functors. We also develop some of the theory of $\mathbf{BO}$-enriched categories. |
| title | What is the universal property of the 2-category of monads? |
| topic | Category Theory 18C15, 18C20, 18D20, 18N10, 18A35 |
| url | https://arxiv.org/abs/2211.02210 |