The categories of corings and coalgebras over a ring are locally countably presentable
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915406954889216 |
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| author | Positselski, Leonid |
| author_facet | Positselski, Leonid |
| contents | For any commutative ring $R$, we show that the categories of $R$-coalgebras and cocommutative $R$-coalgebras are locally $\aleph_1$-presentable, while the categories of $R$-flat $R$-coalgebras are $\aleph_1$-accessible. Similarly, for any associative ring $R$, the category of $R$-corings is locally $\aleph_1$-presentable, while the category of $R$-$R$-bimodule flat $R$-corings is $\aleph_1$-accessible. The cardinality of the ring $R$ can be arbitrarily large. We also discuss $R$-corings with surjective counit and flat kernel. The proofs are straightforward applications of an abstract category-theoretic principle going back to Ulmer. For right or two-sided $R$-module flat $R$-corings, our cardinality estimate for the accessibility rank is not as good. A generalization to comonoid objects in accessible monoidal categories is also considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02928 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The categories of corings and coalgebras over a ring are locally countably presentable Positselski, Leonid Rings and Algebras Category Theory For any commutative ring $R$, we show that the categories of $R$-coalgebras and cocommutative $R$-coalgebras are locally $\aleph_1$-presentable, while the categories of $R$-flat $R$-coalgebras are $\aleph_1$-accessible. Similarly, for any associative ring $R$, the category of $R$-corings is locally $\aleph_1$-presentable, while the category of $R$-$R$-bimodule flat $R$-corings is $\aleph_1$-accessible. The cardinality of the ring $R$ can be arbitrarily large. We also discuss $R$-corings with surjective counit and flat kernel. The proofs are straightforward applications of an abstract category-theoretic principle going back to Ulmer. For right or two-sided $R$-module flat $R$-corings, our cardinality estimate for the accessibility rank is not as good. A generalization to comonoid objects in accessible monoidal categories is also considered. |
| title | The categories of corings and coalgebras over a ring are locally countably presentable |
| topic | Rings and Algebras Category Theory |
| url | https://arxiv.org/abs/2401.02928 |