We functorially associate to each relative –category a simplicial space , 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 is precisely the one corresponding to the localization ; and (ii) that the Rezk nerve functor defines an equivalence from a localization of the –category of relative –categories to the –category of –categories.
"The universality of the Rezk nerve." Algebr. Geom. Topol. 19 (7) 3217 - 3260, 2019. https://doi.org/10.2140/agt.2019.19.3217