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arXiv · 2201.09589

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

Fernando Abellán·Walker H. Stern
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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.

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Fernando Abellán, Walker H. Stern. 2022-01-24. 2-Cartesian fibrations II: A Grothendieck construction for $\infty$-bicategories. https://arxiv.org/abs/2201.09589

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