The universality of the Rezk nerve

TL;DR

This paper introduces the Rezk nerve for relative ∞-categories, establishing its universal properties and showing it induces an equivalence between localized relative ∞-categories and ∞-categories, thus demonstrating its foundational role.

Contribution

It generalizes Rezk's classification diagram to relative ∞-categories and proves its universal properties, linking relative ∞-categories to ∞-categories via the Rezk nerve.

Findings

Rezk nerve's complete Segal space corresponds to localization

Rezk nerve induces an equivalence between localized relative ∞-categories and ∞-categories

Universal properties of the Rezk nerve are established

Abstract

We functorially associate to each relative $\infty$-category $(R,W)$ a simplicial space $N^R_\infty(R,W)$, called its Rezk nerve (a straightforward generalization of Rezk's "classification diagram" construction for relative categories). We prove the following local and global universal properties of this construction: (i) that the complete Segal space generated by the Rezk nerve $N^R_\infty(R,W)$ is precisely the one corresponding to the localization $R[[W^{-1}]]$; and (ii) that the Rezk nerve functor defines an equivalence $RelCat_\infty [[ W_{BK}^{-1} ]] \xrightarrow{\sim} Cat_\infty$ from a localization of the $\infty$-category of relative $\infty$-categories to the $\infty$-category of $\infty$-categories.