The universality of the Rezk nerve

Abstract

We functorially associate to each relative $\infty$–category $\left(\mathcal{ℛ},\mathbf{W}\right)$ a simplicial space $N_{\infty }^R\left(\mathcal{ℛ},\mathbf{W}\right)$, 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 $N_{\infty }^R\left(\mathcal{ℛ},\mathbf{W}\right)$ is precisely the one corresponding to the localization $\mathcal{ℛ}\left[\left[{\mathbf{W}}^{-1}\right]\right]$; and (ii) that the Rezk nerve functor defines an equivalence $\mathcal{ℛ}el\mathcal{C}{at}_{\infty }\left[\left[{\mathbf{W}}_{BK}^{-1}\right]\right]\underset{}{\overset{\sim }{\to }}\mathcal{C}{at}_{\infty }$ from a localization of the $\infty$–category of relative $\infty$–categories to the $\infty$–category of $\infty$–categories.