Higher presentable categories and limits
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914094984986624 |
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| author | Aoki, Ko |
| author_facet | Aoki, Ko |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13503 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher presentable categories and limits Aoki, Ko Category Theory Algebraic Topology Logic 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. |
| title | Higher presentable categories and limits |
| topic | Category Theory Algebraic Topology Logic |
| url | https://arxiv.org/abs/2510.13503 |