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

TL;DR

This paper develops a comprehensive theory of fibrations and 2-(co)limits in $ ext{(} ext{infty} ext{,}2 ext{)}$-categories, connecting abstract higher category theory with model categorical computations and establishing foundational results on their existence and uniqueness.

Contribution

It introduces four types of fibrations encoding variance of diagrams in $ ext{(} ext{infty} ext{,}2 ext{)}$-categories and develops a general theory of 2-(co)limits, including their calculation via weighted homotopy limits.

Findings

Four types of fibrations encode variance of diagrams.

2-(co)limits can be computed as weighted homotopy limits.

Existence and unicity of 2-(co)limits are established.

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.