Bi-accessible and bipresentable 2-categories
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866913970025136128 |
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| author | Di Liberti, Ivan Osmond, Axel |
| author_facet | Di Liberti, Ivan Osmond, Axel |
| contents | We develop a 2-dimensional version of accessibility and presentability compatible with the formalism of flat pseudofunctors. First we give prerequisites on the different notions of 2-dimensional colimits, filteredness and cofinality; in particular we show that sigma-filteredness and bifilteredness are actually equivalent in practice for our purposes. Then, we define bi-accessible and bipresentable 2-categories in terms of bicompact objects and bifiltered bicolimits. We then characterize them as categories of flat pseudofunctors. We also prove a bi-accessible right bi-adjoint functor theorem and deduce a 2-dimensional Gabriel-Ulmer duality relating small bilex 2-categories and finitely bipresentable 2-categories. Finally, we show that 2-categories of pseudo-algebras of bifinitary pseudomonads on Cat are finitely bipresentable, which in particular captures the case of Lex, the 2-category of small lex categories. Invoking the technology of lex-colimits, we prove further that several 2-categories arising in categorical logic (Reg, Ex, Coh, Ext, Adh, Pretop) are also finitely bipresentable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_07046 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bi-accessible and bipresentable 2-categories Di Liberti, Ivan Osmond, Axel Category Theory We develop a 2-dimensional version of accessibility and presentability compatible with the formalism of flat pseudofunctors. First we give prerequisites on the different notions of 2-dimensional colimits, filteredness and cofinality; in particular we show that sigma-filteredness and bifilteredness are actually equivalent in practice for our purposes. Then, we define bi-accessible and bipresentable 2-categories in terms of bicompact objects and bifiltered bicolimits. We then characterize them as categories of flat pseudofunctors. We also prove a bi-accessible right bi-adjoint functor theorem and deduce a 2-dimensional Gabriel-Ulmer duality relating small bilex 2-categories and finitely bipresentable 2-categories. Finally, we show that 2-categories of pseudo-algebras of bifinitary pseudomonads on Cat are finitely bipresentable, which in particular captures the case of Lex, the 2-category of small lex categories. Invoking the technology of lex-colimits, we prove further that several 2-categories arising in categorical logic (Reg, Ex, Coh, Ext, Adh, Pretop) are also finitely bipresentable. |
| title | Bi-accessible and bipresentable 2-categories |
| topic | Category Theory |
| url | https://arxiv.org/abs/2203.07046 |