This page describes some nice properties of the category of cocommutative? coalgebras over a ground ring , in particular details of the proof that it is a complete, cocomplete, lextensive, cartesian closed category with a generating set.
Todd: I have a question I hope someone can answer: is the category of cocommutative coalgebras (let’s say over a field ) locally finitely presentable?
I can say that every object is a filtered colimit (in fact a union) of finite-dimensional subcoalgebras , and it’s tantalizing for me to suppose that for such the functor preserves filtered colimits, although I don’t have a complete proof of that. On the other hand, I can’t find any mention of this purported fact anywhere in the literature, and it looks hard just to write down a finite limit sketch for which is the category of models. I should probably mention that I am fairly ignorant of accessible category theory, in case it’s not already obvious. All help is appreciated.
Daniel Schäppi: There is a paper on this by Hans-E. Porst: On Corings and Comodules, Arch. Math. (Brno) 42 (2006), 419-425. There it is shown that the category of comodules of an R-coalgebra is locally presentable for any commutative ring R, but it is in general not true that it is locally finitely presentable.
Michael Barr, Coalgebras over a commutative ring, J. Alg. 32 (1974), 600–610.