2025/10/15 by Ko Aoki, Aoki, Ko · 2 citations
Computer Science · #Advanced Algebra and Logic #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2510.13503
openalex publication_date 2025/10/15 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28
Stefanich generalized the notion of (locally) presentable (∞, 1)-category to the notion of presentable (∞, n)-category. We give a new description based on the new notion of κ-compactly generated (∞, n)-category, which avoids universe enlargement. Using the new definition, we prove the underlying functor of a morphism between presentable (∞, 2)-categories has a right adjoint. In particular, any presentable (∞, 2)-category has limits. We also prove that this fails drastically when we go higher: The unit presentable (∞, 3)-category, i.e., the category of presentable (∞, 2)-categories, does not have limits. This settles Stefanich's conjecture in the negative.