Free fibrations, lax colimits and Kan extensions for $(\infty,2)$-categories

Abstract

In the first part of this paper we study fibrations of $(\infty,2)$-categories. We give a simple characterization of such fibrations in terms of a certain square being a pullback, and apply this to show that in some cases $(\infty,2)$-categories of functors and partially (op)lax transformations preserve fibrations. We also describe free fibrations of $(\infty,2)$-categories, including in the case where we only ask for (co)cartesian lifts of specified 1- and 2-morphisms in the base, and describe the right adjoint to pullback from fibrations to such partial fibrations along an arbitrary functor. In the second part of the paper we apply these results to study colimits and Kan extensions of $(\infty,2)$-categories. Most notably, we give a fibrational description of both partially (op)lax and weighted (co)limits of $(\infty,2)$-categories and construct partially lax Kan extensions. Among other results, we also include a model-independent version of cofinality for $(\infty,2)$-categories and briefly consider presentable $(\infty,2)$-categories, characterizing them as accessible localizations of presheaves of $\infty$-categories.