Is there a finite abelian category?

Question

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.

Edit:

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.

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Any help is sincerely appreciated.

Answer 1

To put this question off the unanswered-list:

The question got an answer on mathoverflow by Jeremy Rickard:

"Take the category of (at most) countable-dimensional vector spaces over your favourite field. Then take the quotient by the Serre subcategory of finite-dimensional vector spaces. (And take a skeletal subcategory so that it strictly has only two objects.)

Then this is an abelian category with only one non-zero object, whose endomorphism ring is the endomorphism ring of a countable-dimensional vector space, localized at the set of endomorphisms with finite-dimensional kernel and cokernel."