Recognizing quasi-categorical limits and colimits in homotopy coherent nerves

TL;DR

This paper proves that a broad class of quasi-categories are complete with limits for all simplicial sets, by characterizing limit cones via homotopy coherent nerves and pseudo homotopy limits, thus generalizing known results.

Contribution

It provides a general theorem characterizing limit cones in quasi-categories as homotopy coherent nerves, extending the understanding of limits in higher category theory.

Findings

Quasi-categories are complete with limits indexed by all simplicial sets.

Limit cones can be modeled by pseudo homotopy limit cones at the point-set level.

The core of an ∞-cosmos admits weighted homotopy limits for all flexible weights.

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.