The universality of the Rezk nerve

Mazel-Gee, Aaron¹

¹ Department of Mathematics, University of Southern California, Los Angeles, CA, United States

Algebraic and Geometric Topology, Tome 19 (2019) no. 7, pp. 3217-3260

Résumé

We functorially associate to each relative ∞–category (ℛ,W) a simplicial space N∞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∞R(ℛ,W) is precisely the one corresponding to the localization ℛ[[W−1]]; and (ii) that the Rezk nerve functor defines an equivalence ℛelCat∞[[WBK−1]] →∼Cat∞ from a localization of the ∞–category of relative ∞–categories to the ∞–category of ∞–categories.

Détail

DOI : 10.2140/agt.2019.19.3217

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