A relative 2-nerve

Abstract

We introduce a 2–categorical variant of Lurie’s relative nerve functor. We prove that it defines a right Quillen equivalence which, upon passage to ∞–categorical localizations, corresponds to Lurie’s scaled unstraightening equivalence. In this ∞–bicategorical context, the relative 2–nerve provides a computationally tractable model for the Grothendieck construction which becomes equivalent, via an explicit comparison map, to Lurie’s relative nerve when restricted to 1–categories.