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

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.