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 –category (,W) a simplicial space NR(,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 NR(,W) is precisely the one corresponding to the localization [[W1]]; and (ii) that the Rezk nerve functor defines an equivalence elCat[[WBK1]] Cat from a localization of the –category of relative –categories to the –category of –categories.

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