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What is an example of a thin abelian (or a thin poset) category? Or perhaps such categories cannot exist?

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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.

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