2024/01/05 by Positselski, Leonid
#Category Theory (math.CT) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2401.02928
For any commutative ring R, we show that the categories of R-coalgebras and cocommutative R-coalgebras are locally ℵ1-presentable, while the categories of R-flat R-coalgebras are ℵ1-accessible. Similarly, for any associative ring R, the category of R-corings is locally ℵ1-presentable, while the category of R-R-bimodule flat R-corings is ℵ1-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.