TITLE: A thin abelian category
QUESTION [5 upvotes]: What is an example of a thin abelian (or a thin poset) category? Or perhaps such categories cannot exist?

REPLY [4 votes]: They are the same as complete directed graphs, so they all are equivalent to trivial abelian category.
Since an abelian category admits a zero arrow between any two objects, so that there is at least one arrow between any two objects, and a zero arrow as an endomorphism on any object, every thin abelian category looks like a complete directed graph. Conversely, every complete directed graph admits a trivial structure as an abelian category.