Recognizing quasi-categorical limits and colimits in homotopy coherent nerves
| Autor: | Riehl, Emily, Verity, Dominic |
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| Rok vydání: | 2018 |
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| Druh dokumentu: | Working Paper |
| Popis: | In this paper we prove that various quasi-categories whose objects are $\infty$-categories in a very general sense are complete: admitting limits indexed by all simplicial sets. This result and others of a similar flavor follow from a general theorem in which we characterize the data that is required to define a limit cone in a quasi-category constructed as a homotopy coherent nerve. Since all quasi-categories arise this way up to equivalence, this analysis covers the general case. Namely, we show that quasi-categorical limit cones may be modeled at the point-set level by pseudo homotopy limit cones, whose shape is governed by the weight for pseudo limits over a homotopy coherent diagram but with the defining universal property up to equivalence, rather than isomorphism, of mapping spaces. Our applications follow from the fact that the $(\infty,1)$-categorical core of an $\infty$-cosmos admits weighted homotopy limits for all flexible weights, which includes in particular the weight for pseudo cones. Comment: 53 pages; a continuation of the program developed in the papers arXiv:1306.5144, arXiv:1310.8279, arXiv:1401.6247, arXiv:1506.05500, arXiv:1507.01460, arXiv:1706.10023 and a precursor to arXiv:1808.09835, as summarized in arXiv:1608.05314; v3 peer-reviewed, with a referee-suggested proof of 6.2.7 |
| Databáze: | arXiv |
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