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arXiv · 2401.02928

The categories of corings and coalgebras over a ring are locally countably presentable

Leonid Positselski
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Abstract

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.

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Leonid Positselski. 2024-01-05. The categories of corings and coalgebras over a ring are locally countably presentable. https://doi.org/10.1007/s10474-025-01538-y

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