Groupoidal and truncated $n$-quasi-categories
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914821002231808 |
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| author | Brittes, Victor |
| author_facet | Brittes, Victor |
| contents | We define groupoidal and $(n+k)$-truncated $n$-quasi-categories, which are the translation to the world of $n$-quasi-categories of groupoidal and truncated $(\infty, n)$-$Θ$-spaces defined by Rezk. We show that these objects are the fibrant objects of model structures on the category of presheaves on $Θ_n$ obtained by localisation of Ara's model structure for $n$-quasi-categories. Furthermore, we prove that the inclusion $Δ\to Θ_n$ induces a Quillen equivalence between the model structure for groupoidal (resp. and $n$-truncated) $n$-quasi-categories and the Kan-Quillen model structure for spaces (resp. homotopy $n$-types) on simplicial sets. To get to these results, we also construct a cylinder object for $n$-quasi-categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01490 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Groupoidal and truncated $n$-quasi-categories Brittes, Victor Category Theory Algebraic Topology 18N20 (Primary) 18N40, 18N55, 18N65, 55P15 (Secondary) We define groupoidal and $(n+k)$-truncated $n$-quasi-categories, which are the translation to the world of $n$-quasi-categories of groupoidal and truncated $(\infty, n)$-$Θ$-spaces defined by Rezk. We show that these objects are the fibrant objects of model structures on the category of presheaves on $Θ_n$ obtained by localisation of Ara's model structure for $n$-quasi-categories. Furthermore, we prove that the inclusion $Δ\to Θ_n$ induces a Quillen equivalence between the model structure for groupoidal (resp. and $n$-truncated) $n$-quasi-categories and the Kan-Quillen model structure for spaces (resp. homotopy $n$-types) on simplicial sets. To get to these results, we also construct a cylinder object for $n$-quasi-categories. |
| title | Groupoidal and truncated $n$-quasi-categories |
| topic | Category Theory Algebraic Topology 18N20 (Primary) 18N40, 18N55, 18N65, 55P15 (Secondary) |
| url | https://arxiv.org/abs/2406.01490 |