Nerves of generalized multicategories
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912962736816128 |
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| author | Fujii, Soichiro Lack, Stephen |
| author_facet | Fujii, Soichiro Lack, Stephen |
| contents | For any category ${\mathcal E}$ and monad $T$ thereon, we introduce the notion of $T$-simplicial object in ${\mathcal E}$. Any $T$-category in the sense of Burroni induces a $T$-simplicial object as its nerve. This nerve construction defines a fully faithful functor from the category $\mathbf{Cat}_T({\mathcal E})$ of $T$-categories to the category $s_T({\mathcal E})$ of $T$-simplicial objects, whose essential image is characterized by a simple condition. We show that the category $s_T({\mathcal E})$ is enriched over the category of simplicial sets, and that this induces the usual 2-category structure on $\mathbf{Cat}_T({\mathcal E})$. We also study enriched limits and colimits in $s_T({\mathcal E})$ and $\mathbf{Cat}_T({\mathcal E})$, and show that if ${\mathcal E}$ is locally finitely presentable and $T$ is finitary, then $\mathbf{Cat}_T({\mathcal E})$ is locally finitely presentable as a 2-category and $s_T({\mathcal E})$ is locally finitely presentable as a simplicially-enriched category. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05232 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nerves of generalized multicategories Fujii, Soichiro Lack, Stephen Category Theory 18M65, 18N50, 18C35, 18N10, 18D20 For any category ${\mathcal E}$ and monad $T$ thereon, we introduce the notion of $T$-simplicial object in ${\mathcal E}$. Any $T$-category in the sense of Burroni induces a $T$-simplicial object as its nerve. This nerve construction defines a fully faithful functor from the category $\mathbf{Cat}_T({\mathcal E})$ of $T$-categories to the category $s_T({\mathcal E})$ of $T$-simplicial objects, whose essential image is characterized by a simple condition. We show that the category $s_T({\mathcal E})$ is enriched over the category of simplicial sets, and that this induces the usual 2-category structure on $\mathbf{Cat}_T({\mathcal E})$. We also study enriched limits and colimits in $s_T({\mathcal E})$ and $\mathbf{Cat}_T({\mathcal E})$, and show that if ${\mathcal E}$ is locally finitely presentable and $T$ is finitary, then $\mathbf{Cat}_T({\mathcal E})$ is locally finitely presentable as a 2-category and $s_T({\mathcal E})$ is locally finitely presentable as a simplicially-enriched category. |
| title | Nerves of generalized multicategories |
| topic | Category Theory 18M65, 18N50, 18C35, 18N10, 18D20 |
| url | https://arxiv.org/abs/2512.05232 |