Higher presentable categories and limits

Abstract

Stefanich generalized the notion of (locally) presentable $(\infty ,1)$-category to the notion of presentable $(\infty ,n)$-category. We give a new description based on the new notion of $\hat{I}º$-compactly generated $(\infty ,n)$-category, which avoids universe enlargement. Using the new definition, we prove the underlying functor of a morphism between presentable $(\infty ,2)$-categories has a right adjoint. In particular, any presentable $(\infty ,2)$-category has limits. We also prove that this fails drastically when we go higher: The unit presentable $(\infty ,3)$-category, i.e., the category of presentable $(\infty ,2)$-categories, does not have limits. This settles Stefanich's conjecture in the negative.

14 pages