Lax functorialities of the comma construction for $ω$-categories
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908760690130944 |
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| author | Ara, Dimitri Guetta, Léonard |
| author_facet | Ara, Dimitri Guetta, Léonard |
| contents | Motivated by the Grothendieck construction, we study the functorialities of the comma construction for strict $ω$-categories. To state the most general functorialities, we use the language of Gray $ω$-categories, that is, categories enriched in the category of strict $ω$-categories endowed with the oplax Gray tensor product. Our main result is that the comma construction of strict $ω$-categories defines a Gray $ω$-functor, that is, a morphism of Gray $ω$-categories. To makes sense of this statement, we prove that slices of Gray $ω$-categories exist. Coming back to the Grothendieck construction, we propose a definition in terms of the comma construction and, as a consequence, we get that the Grothendieck construction of strict $ω$-categories defines a Gray $ω$-functor. Finally, as a by-product, we get a notion of Grothendieck construction for Gray $ω$-functors, which we plan to investigate in future work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08832 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lax functorialities of the comma construction for $ω$-categories Ara, Dimitri Guetta, Léonard Category Theory 18A25, 18D20, 18N20, 18N30 Motivated by the Grothendieck construction, we study the functorialities of the comma construction for strict $ω$-categories. To state the most general functorialities, we use the language of Gray $ω$-categories, that is, categories enriched in the category of strict $ω$-categories endowed with the oplax Gray tensor product. Our main result is that the comma construction of strict $ω$-categories defines a Gray $ω$-functor, that is, a morphism of Gray $ω$-categories. To makes sense of this statement, we prove that slices of Gray $ω$-categories exist. Coming back to the Grothendieck construction, we propose a definition in terms of the comma construction and, as a consequence, we get that the Grothendieck construction of strict $ω$-categories defines a Gray $ω$-functor. Finally, as a by-product, we get a notion of Grothendieck construction for Gray $ω$-functors, which we plan to investigate in future work. |
| title | Lax functorialities of the comma construction for $ω$-categories |
| topic | Category Theory 18A25, 18D20, 18N20, 18N30 |
| url | https://arxiv.org/abs/2503.08832 |