The universality of the Rezk nerve

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.