Universality of categories of coalgebras (Q409255)

A set functor \(F\) is accessible if it is a quotient functor of a coproduct of a set of hom-functors. \(F\) is called narrow if it preserves unions of finitely many non-void sets. The following four main results are proved in the paper: (1) Under GCH, the category \(\mathrm{Coalg}F\) is universal if and only if the set functor is not narrow. (2) Under GCH, for any non-accessible functor \(F\), the category \(\mathrm{Coalg}F\) has a large discrete category as a full subcategory. (3) For any non-accessible intersection-preserving functor \(F\), the category \(\mathrm{Coalg}F\) has a large discrete full subcategory. (4) The non-universal category \(\mathrm{Coalg}\beta\) (\(\beta\) being a Stone-Čech functor) has a large discrete full subcategory.