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Mathematics > Algebraic Topology

arXiv:2201.09589 (math)
[Submitted on 24 Jan 2022 (v1), last revised 13 Apr 2023 (this version, v3)]

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

Authors: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.
Comments: Comments welcome! v2: This version only includes the Grothendieck construction. The results regarding cofinality are now appearing in a separate paper
Subjects: Algebraic Topology (math.AT); Category Theory (math.CT)
Cite as: arXiv:2201.09589 [math.AT]
  (or arXiv:2201.09589v3 [math.AT] for this version)
  https://doi.org/10.48550/arXiv.2201.09589

Submission history

From: Fernando Abellán García [view email]
[v1] Mon, 24 Jan 2022 11:02:27 UTC (88 KB)
[v2] Wed, 2 Mar 2022 14:14:16 UTC (88 KB)
[v3] Thu, 13 Apr 2023 09:28:25 UTC (68 KB)
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