2-Cartesian fibrations II: A Grothendieck construction for $\infty$-bicategories

TL;DR

This paper develops a Grothendieck construction for fibred $ abla$-bicategories, establishing an equivalence between 2-Cartesian fibrations over a scaled simplicial set and contravariant functors into the $ abla$-bicategory of $ abla$-bicategories.

Contribution

It extends Lurie's straightening-unstraightening adjunction to the setting of $ abla$-bicategories, providing a new equivalence and a comparison to existing bicategorical Grothendieck constructions.

Findings

Constructs a 2-categorical straightening-unstraightening adjunction.

Establishes an equivalence between 2-Cartesian fibrations and contravariant functors.

Provides a relative nerve construction and compares to existing frameworks.

Abstract

In this work, we conclude our study of fibred $\infty$-bicategories by providing a Grothendieck construction in this setting. Given a scaled simplicial set $S$ (which need not be fibrant) we construct a 2-categorical version of Lurie's straightening-unstraightening adjunction, thereby furnishing an equivalence between the $\infty$-bicategory of 2-Cartesian fibrations over $S$ and the $\infty$-bicategory of contravariant functors $S^{\operatorname{op}} \to \mathbb{B}\mathbf{\!}\operatorname{icat}_\infty$ with values in the $\infty$-bicategory of $\infty$-bicategories. We provide a relative nerve construction in the case where the base is a 2-category, and use this to prove a comparison to existing bicategorical Grothendieck constructions.