On Cofiltered Limits of $\infty$-Categories and Adjunctions

Abstract

We prove that, under mild assumptions, the limit of a cofiltered diagram of $\infty$-categories is a reflective (resp. coreflective) localization of its oplax (resp. lax) limit. This gives rise to a number of exceptional 'stability' results for categorical properties under such limits, in particular one for adjunctions. As an application, we recover a push-pull formula describing certain filtered colimits in the $\infty$-category ${\mathrm{Pr}}^L$ of presentable $\infty$-categories.

13 pages