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Applied Categorical Structures

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06-03-2020

Recognizing Quasi-Categorical Limits and Colimits in Homotopy Coherent Nerves

Authors: Emily Riehl, Dominic Verity

Published in: Applied Categorical Structures | Issue 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.

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Known \(\infty \)-cosmoi of \((\infty ,n)\)-categories include \(\theta _n\)-spaces, iterated complete Segal spaces, n-complicial sets, and n-quasi-categories.

Here we show that the codomain of the comparison map in (4.1.9) is a quasi-category by applying Proposition 4.1.5 in the \(\infty \)-cosmos of quasi-categories.

The simplicial computads are the cofibrant objects [12, §16.2] in the model structure on simplicial categories due to Bergner [2].

For bookkeeping reasons it is convenient to adopt the convention that atomic arrows are not identities, though in a simplicial computad the identities will also admit no non-trivial factorisations. With this convention, an identity arrow factors uniquely as an empty composite of atomic arrows.

For more details about the Leibniz or “pushout-product” construction see [13, §4].

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Title: Recognizing Quasi-Categorical Limits and Colimits in Homotopy Coherent Nerves
Authors: Emily Riehl
Dominic Verity
Publication date: 06-03-2020
Publisher: Springer Netherlands
Published in: Applied Categorical Structures / Issue 4/2020
Print ISSN: 0927-2852
Electronic ISSN: 1572-9095
DOI: https://doi.org/10.1007/s10485-020-09594-x

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