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.

LaTeX 2e with xy-pic; 25 pages, 6 commutative diagrams; v.4: new Section 2 inserted; v.5: two paragraphs moved from the proof of Theorem 3.2 to the proof of Theorem 2.1, details added in the proof of Lemma 3.1, explanations added in the proof of Theorem 2.1; v.6: several misprints corrected; v.7: small things corrected, a paragraph inserted at the end of the proof of Lemma 3.1

References in corpus (3)

• Flat comodules and contramodules as directed colimits, and cotorsion periodicity
• Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity
• Resolutions as directed colimits