A relative 2-nerve

TL;DR

This paper introduces a 2-categorical version of Lurie's relative nerve functor, establishing a Quillen equivalence and providing a computational model for the Grothendieck construction within an $bicategorical framework.

Contribution

It develops a 2-categorical relative nerve, proves its equivalence to Lurie's scaled unstraightening, and offers a tractable model for the Grothendieck construction.

Findings

Establishes a right Quillen equivalence for the 2-relative nerve.

Shows the 2-relative nerve corresponds to Lurie's scaled unstraightening.

Provides an explicit comparison map to Lurie's relative nerve for 1-categories.

Abstract

In this work, 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.