Higher presentable categories and limits
Ko Aoki
TL;DR
This work develops a universe-free framework for presentable $n$-categories by introducing $\kappa$-compactly generated structures, defining $0\mathsf{Pr}^{\kappa}=\mathsf{Ani}$ and $(n+1)\mathsf{Pr}^{\kappa}=\mathrm{Mod}_{n\mathsf{Pr}^{\kappa}}\mathsf{Pr}^{\kappa}$, and forming $n\mathsf{Pr}=\varinjlim_{\kappa}n\mathsf{Pr}^{\kappa}$. It proves a positive adjoint-functor theorem for presentable $(\infty,2)$-categories: the underlying functor of any morphism has a right adjoint, hence these categories possess limits, while showing that the unit presentable $(\infty,3)$-category of presentable $(\infty,2)$-categories does not admit limits, answering Stefanich’s conjecture in the negative. The paper also develops enriched Ind-categories to model limits and conducts a detailed analysis of dominant morphisms, their complexity, and global limits, culminating in explicit nonpresentable pullbacks and mapping categories at higher levels. The results illuminate the asymmetry of adjoints and limits in higher category theory, provide a robust, enlargement-free construction of presentable $n$-categories, and reveal intrinsic higher-dimensional pathologies absent at the $(\infty,2)$-level.
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 $κ$-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.