Cartesian fibrations of (∞,2)–categories

Gagna, Andrea; Harpaz, Yonatan; Lanari, Edoardo

doi:10.2140/agt.2024.24.4731

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Cartesian fibrations of $(\infty,2)$–categories

Andrea Gagna, Yonatan Harpaz and Edoardo Lanari

Algebraic & Geometric Topology 24 (2024) 4731–4778

DOI: 10.2140/agt.2024.24.4731

Abstract

We introduce four variance flavors of (co)cartesian fibrations of $\infty$ –bicategories with $\infty$ –bicategorical fibers, in the framework of scaled simplicial sets. Given a map $p : ℰ \to ℬ$ of $\infty$ –bicategories, we define $p$ –(co)cartesian arrows and inner/outer triangles by means of lifting properties against $p$ , leading to a notion of $2$ –inner/outer (co)cartesian fibrations as those maps with enough (co)cartesian lifts for arrows and enough inner/outer lifts for triangles, together with a compatibility property with respect to whiskerings in the outer case. By doing so, we also recover in particular the case of $\infty$ –bicategories fibered in $\infty$ –categories studied in previous work. We also prove that equivalences of such fibrations can be tested fiberwise. As a motivating example, we show that the domain projection $d : {Fun}^{g r} (Δ^{1}, 𝒞) \to 𝒞$ is a prototypical example of a $2$ –outer cartesian fibration, where ${Fun}^{g r} (X, Y)$ denotes the $\infty$ –bicategory of functors, lax natural transformations and modifications. We then define $2$ –inner and $2$ –outer flavors of (co)cartesian fibrations of categories enriched in $\infty$ –categories, and we show that a fibration $p : ℰ \to ℬ$ of such enriched categories is a (co)cartesian $2$ –inner/outer fibration if and only if the corresponding map $N^{sc} (p) : N^{sc} ℰ \to N^{sc} ℬ$ is a fibration of this type between $\infty$ –bicategories.

Keywords

fibrations, higher categories, bicategories

Mathematical Subject Classification

Primary: 18N65, 18N50, 55U35

References

Publication

Received: 27 July 2021

Revised: 5 February 2023

Accepted: 30 September 2023

Published: 27 December 2024

Authors

Andrea Gagna
Institute of Mathematics Czech Academy of Sciences Prague Czech Republic
https://sites.google.com/view/andreagagna/home
Yonatan Harpaz
Institut Galilée Université Paris 13 Villeta-neuse France
https://www.math.univ-paris13.fr/~harpaz
Edoardo Lanari
Institute of Mathematics Czech Academy of Sciences Prague Czech Republic
https://edolana.github.io/

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Volume 24, issue 9 (2024)