The universality of the Rezk nerve

Mazel-Gee, Aaron

doi:10.2140/agt.2019.19.3217

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The universality of the Rezk nerve

Aaron Mazel-Gee

Algebraic & Geometric Topology 19 (2019) 3217–3260

DOI: 10.2140/agt.2019.19.3217

Abstract

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

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Keywords

$\infty$–category, relative $\infty$–category, localization, Rezk nerve, classification diagram

Mathematical Subject Classification 2010

Primary: 18A05, 55U35

References

Publication

Received: 8 December 2015

Revised: 13 January 2019

Accepted: 29 January 2019

Published: 17 December 2019

Authors

Aaron Mazel-Gee
Department of Mathematics University of Southern California Los Angeles, CA United States

Volume 19, issue 7 (2019)