Volume 19, issue 7 (2019)

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The universality of the Rezk nerve

Aaron Mazel-Gee

Algebraic & Geometric Topology 19 (2019) 3217–3260
Abstract

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

Keywords
$\infty$–category, relative $\infty$–category, localization, Rezk nerve, classification diagram
Mathematical Subject Classification 2010
Primary: 18A05, 55U35