Enriched quasi-categories and the templicial homotopy coherent nerve
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866918025986310144 |
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| author | Lowen, Wendy Mertens, Arne |
| author_facet | Lowen, Wendy Mertens, Arne |
| contents | We lay the foundations for a theory of quasi-categories in a monoidal category $\mathcal{V}$ replacing $\mathrm{Set}$, aimed at realising weak enrichment in the category $S\mathcal{V}$ of simplicial objects in $\mathcal{V}$. To accomodate non-cartesian monoidal products, we make use of an ambient category $S_{\otimes}\mathcal{V}$ of templicial - or 'tensor-simplicial' - objects in $\mathcal{V}$, which are certain colax monoidal functors following Leinster. Inspired by the description of the categorification functor due to Dugger and Spivak, we construct a templicial analogue of the homotopy coherent nerve functor which goes from $S\mathcal{V}$-enriched categories to templicial objects. We show that an $S\mathcal{V}$-enriched category whose underlying simplicial category is locally Kan, is turned into a quasi-category in $\mathcal{V}$ by this nerve functor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_02484 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Enriched quasi-categories and the templicial homotopy coherent nerve Lowen, Wendy Mertens, Arne Category Theory 18N60, 18D20 (Primary), 18N50, 18M05 (Secondary) We lay the foundations for a theory of quasi-categories in a monoidal category $\mathcal{V}$ replacing $\mathrm{Set}$, aimed at realising weak enrichment in the category $S\mathcal{V}$ of simplicial objects in $\mathcal{V}$. To accomodate non-cartesian monoidal products, we make use of an ambient category $S_{\otimes}\mathcal{V}$ of templicial - or 'tensor-simplicial' - objects in $\mathcal{V}$, which are certain colax monoidal functors following Leinster. Inspired by the description of the categorification functor due to Dugger and Spivak, we construct a templicial analogue of the homotopy coherent nerve functor which goes from $S\mathcal{V}$-enriched categories to templicial objects. We show that an $S\mathcal{V}$-enriched category whose underlying simplicial category is locally Kan, is turned into a quasi-category in $\mathcal{V}$ by this nerve functor. |
| title | Enriched quasi-categories and the templicial homotopy coherent nerve |
| topic | Category Theory 18N60, 18D20 (Primary), 18N50, 18M05 (Secondary) |
| url | https://arxiv.org/abs/2302.02484 |