Hammocks and fractions in relative $\infty$-categories

TL;DR

This paper develops a homotopy theory framework for $ $-categories enriched in simplicial spaces, establishing an equivalence with ordinary $ $-categories and providing explicit descriptions of hom-spaces via hammock localization.

Contribution

It introduces a model for enriched $ $-categories via hammock localization and generalizes key results to describe hom-spaces explicitly in this setting.

Findings

Homotopy theory of $sS$-enriched $ $-categories presents the $ $-category of $ $-categories.

Hammock localization provides a rich source of examples of $sS$-enriched $ $-categories.

Explicit descriptions of hom-spaces are obtained under a homotopical three-arrow calculus.

Abstract

We study the *homotopy theory* of $\infty$-categories enriched in the $\infty$-category $sS$ of simplicial spaces. That is, we consider $sS$-enriched $\infty$-categories as presentations of ordinary $\infty$-categories by means of a "local" geometric realization functor $Cat_{sS} \to Cat_\infty$, and we prove that their homotopy theory presents the $\infty$-category of $\infty$-categories, i.e. that this functor induces an equivalence $Cat_{sS} [[ W_{DK}^{-1} ]] \xrightarrow{\sim} Cat_\infty$ from a localization of the $\infty$-category of $sS$-enriched $\infty$-categories. Following Dwyer--Kan, we define a *hammock localization* functor from relative $\infty$-categories to $sS$-enriched $\infty$-categories, thus providing a rich source of examples of $sS$-enriched $\infty$-categories. Simultaneously unpacking and generalizing one of their key results, we prove that given a relative $\infty$-category admitting a *homotopical three-arrow calculus*, one can explicitly describe the hom-spaces in the $\infty$-category presented by its hammock localization in a much more explicit and accessible way. As an application of this framework, we give sufficient conditions for the Rezk nerve of a relative $\infty$-category to be a (complete) Segal space, generalizing joint work with Low.