Is there a non-discrete abelian category which has only finitely many objects?
Just out of curiosity I am wondering if such an abelian category exists, while the usual examples of abelian categories that I know contain infinitely many objects.
If the finiteness is too strong a condition, then per chance one might ask
Is there an abelian category whose objects form a set?
I have no idea whether such a small abelian category exists; I am not even sure if the set of all abelian groups exists.
Thanks to @Oskar, this question already has an answer here. But I cannot mark this question as duplicate, as that is a math overflow question.
As a brief summary, @Jeremy Rickard gave an answer there: there is an abelian category with two objects, which is any skeletal category of the quotient category of the category of countable dimensional vector-spaces by the Serre subcategory of finite-dimensional vector-spaces.
Any help is sincerely appreciated.