Fibrations and lax Limits of (∞,2)-Categories

We study four types of (co)cartesian fibrations of $\infty$-bicategories over a given base $B$, and prove that they encode the four variance flavors of $B$-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. Next, we extend this to $(\infty,2)$-category valued diagrams and the corresponding fibrations, and we provide motivating examples. We end by discussing a notion of cofinality appropriate to this setting and use it to deduce the unicity of 2-(co)limits, once they exist.

This is joint work with A.Gagna and Y.Harpaz.