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 $\infty$–categorical localizations, corresponds to Lurie’s scaled unstraightening equivalence. In this $\infty$–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.