Free fibrations, lax colimits and Kan extensions for $(\infty,2)$-categories
Fernando Abellán, Rune Haugseng, Louis Martini
TL;DR
The paper develops a comprehensive, model-independent framework for fibrations in $( ablafty{ }{2})$-categories, beginning with a pullback characterization of $(0,1)$-fibrations and extending to decorated and partial fibrations. It then constructs free and decorated partial fibrations, analyzes pushforwards/pullbacks, and establishes straightening/unnormalized straightening results that underlie partially lax Kan extensions and weighted (co)limits in $( ablafty{ }{2})$-categories. By connecting these fibrational descriptions to lax/oplax transformations, lax slices, and decorated Gray tensor products, the work gives fibrational descriptions of partially lax and weighted (co)limits, along with a model-independent cofinality theory and a presentation of presentable $( ablafty{ }{2})$-categories as localizations of presheaves. The results yield a robust toolkit for accessing Kan extensions, cofinality, and (weighted) (co)limits in higher category theory, enabling structured manipulations of (decorated) fibrations and their straightening. Overall, the paper advances the systematic handling of (co)limits and Kan extensions in $( ablafty{ }{2})$-categories via fibrational and decorated-fibration techniques with broad conceptual and technical impact.
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.