On the structure of simplicial categories associated to quasi-categories

TL;DR

This paper investigates the structure of simplicial categories derived from quasi-categories, revealing fillability of certain horns, coskeletality of hom-spaces, and connections to classical categories, advancing understanding of (infinity,1)-categories.

Contribution

It characterizes the hom-spaces of simplicial categories from quasi-categories, proves their 3-coskeletality, and relates these structures to classical categories and known cofibrant replacements.

Findings

Hom-spaces of CX from quasi-categories can fill all 2,1-horns.

Hom-spaces of CX are 3-coskeletal for any simplicial set X.

When X is an ordinary category's nerve, CX matches the standard cofibrant simplicial resolution.

Abstract

The homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint C are important to the study of (infinity,1)-categories because they provide a means for comparing two models of their respective homotopy theories, giving a Quillen equivalence between the model structures for quasi-categories and simplicial categories. The functor C also gives a cofibrant replacement for ordinary categories, regarded as trivial simplicial categories. However, the hom-spaces of the simplicial category CX arising from a quasi-category X are not well understood. We show that when X is a quasi-category, all 2,1-horns in the hom-spaces of its simplicial category can be filled. We prove, unexpectedly, that for any simplicial set X, the hom-spaces of CX are 3-coskeletal. We characterize the quasi-categories whose simplicial categories are locally quasi, finding explicit examples of 3-dimensional horns that cannot be filled in all other cases. Finally, we show that when X is the nerve of an ordinary category, CX is isomorphic to the simplicial category obtained from the standard free simplicial resolution, showing that the two known cofibrant "simplicial thickenings" of ordinary categories coincide, and furthermore its hom-spaces are 2-coskeletal.