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

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 $S^{\mathrm{op}}\to \mathbb{B}{\mathrm{icat}}_{\infty }$ 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.

Comments welcome! v2: This version only includes the Grothendieck construction. The results regarding cofinality are now appearing in a separate paper