Recognizing quasi-categorical limits and colimits in homotopy coherent nerves

Emily Riehl*, Dominic Verity

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


In this paper we prove that various quasi-categories whose objects are ∞-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 (∞, 1) -categorical core of an ∞-cosmos admits weighted homotopy limits for all flexible weights, which includes in particular the weight for pseudo cones.

Original languageEnglish
Pages (from-to)669-716
Number of pages48
JournalApplied Categorical Structures
Issue number4
Publication statusPublished - Aug 2020


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