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Fibrations and lax limits of $(\infty,2)$-categories

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arxiv 2012.04537 v2 pith:Y4CV33YR submitted 2020-12-08 math.AT math.CT

Andrea Gagna , Yonatan Harpaz , Edoardo Lanari This is my paper · ORCID
classification math.ATmath.CT
keywords inftylimitsbicategorycategoriesdiagramsfibrationsfourlimit
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abstract

We study four types of (co)cartesian fibrations of $\infty$-bicategories over a given base $\mathcal{B}$, and prove that they encode the four variance flavors of $\mathcal{B}$-indexed diagrams of $\infty$-categories. We then use this machinery to set up a general theory of 2-(co)limits for diagrams valued in an $\infty$-bicategory, capable of expressing lax, weighted and pseudo limits. When the $\infty$-bicategory at hand arises from a model category tensored over marked simplicial sets, we show that this notion of 2-(co)limit can be calculated as a suitable form of a weighted homotopy limit on the model categorical level, thus showing in particular the existence of these 2-(co)limits in a wide range of examples. We finish by discussing a notion of cofinality appropriate to this setting and use it to deduce the unicity of 2-(co)limits, once exist.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Colimits in Oriented Category Theory

    math.AT 2026-08 conditional novelty 7.0 of 10

    Oriented colimits generalize lax colimits and the Gray tensor product and yield a Gray-enriched straightening equivalence between presheaves and cocartesian fibrations of (∞,∞)-categories.

  2. Lectures on bar and cobar

    math.AT 2025-07 conditional novelty 7.0 of 10

    The author unifies classical and derived bar and cobar constructions via twisted arrow categories of infinity-operads, giving new existence proofs and recovering classical comparison maps.

  3. Fibrations in Oriented Category Theory

    math.AT 2026-07 conditional novelty 6.0 of 10

    The paper establishes several equivalent characterizations of fibrations of (∞,∞)-categories, shows the category of fibrations forms an oriented category, and constructs free, universal, and Grothendieck-construction ...

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