A double $(\infty,1)$-categorical nerve for double categories
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arXiv
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866913322648993792 |
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| author | Moser, Lyne |
| author_facet | Moser, Lyne |
| contents | We construct a nerve from double categories into double $(\infty,1)$-categories and show that it gives a right Quillen and homotopically fully faithful functor between the model structure for weakly horizontally invariant double categories and the model structure on bisimplicial spaces for double $(\infty,1)$-categories seen as double Segal objects in spaces complete in the horizontal direction. We then restrict the nerve along a homotopical horizontal embedding of 2-categories into double categories, and show that it gives a right Quillen and homotopically fully faithful functor between Lack's model structure for 2-categories and the model structure for 2-fold complete Segal spaces. We further show that Lack's model structure is right-induced along this nerve from the model structure for 2-fold complete Segal spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_01848 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A double $(\infty,1)$-categorical nerve for double categories Moser, Lyne Algebraic Topology Category Theory 18N10, 18N65, 55U35 We construct a nerve from double categories into double $(\infty,1)$-categories and show that it gives a right Quillen and homotopically fully faithful functor between the model structure for weakly horizontally invariant double categories and the model structure on bisimplicial spaces for double $(\infty,1)$-categories seen as double Segal objects in spaces complete in the horizontal direction. We then restrict the nerve along a homotopical horizontal embedding of 2-categories into double categories, and show that it gives a right Quillen and homotopically fully faithful functor between Lack's model structure for 2-categories and the model structure for 2-fold complete Segal spaces. We further show that Lack's model structure is right-induced along this nerve from the model structure for 2-fold complete Segal spaces. |
| title | A double $(\infty,1)$-categorical nerve for double categories |
| topic | Algebraic Topology Category Theory 18N10, 18N65, 55U35 |
| url | https://arxiv.org/abs/2007.01848 |