Cocompactness and Presentability

Abstract

We give a short proof that $κ$-cocompact objects in a presentable category are subterminal. As our main result, we extend this to the setting of presentable $\infty$-categories. A consequence is that an $\infty$-category $\mathcal{C}$ such that both $\mathcal{C}$ and $\mathcal{C}^\mathsf{op}$ are presentable is a small complete lattice, extending a classical theorem of Gabriel-Ulmer. Along the way, we prove a nilpotence result for phantom maps in general pointed presentable $\infty$-categories. Additionally, we show that a strengthening of our main result is equivalent to the existence of a proper class of measurable cardinals.