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Applied Categorical Structures

Erschienen in:

06.03.2020

Recognizing Quasi-Categorical Limits and Colimits in Homotopy Coherent Nerves

verfasst von: Emily Riehl, Dominic Verity

Erschienen in: Applied Categorical Structures | Ausgabe 4/2020

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Abstract

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.

Vorheriger Artikel Exact Filters and Joins of Closed Sublocales

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Titel: Recognizing Quasi-Categorical Limits and Colimits in Homotopy Coherent Nerves
verfasst von: Emily Riehl
Dominic Verity
Publikationsdatum: 06.03.2020
Verlag: Springer Netherlands
Erschienen in: Applied Categorical Structures / Ausgabe 4/2020
Print ISSN: 0927-2852
Elektronische ISSN: 1572-9095
DOI: https://doi.org/10.1007/s10485-020-09594-x

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