In many categories (e.g. Set, Top, Grp) one can show that the epimorphisms are precisely the surjective morphisms. This is not true in the category of rings, for instance. Is there anything like a general criterion for when an epimorphism is a surjection? I’m only aware of the result that adding a finiteness condition on the morphisms makes the statement true for rings.
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See https://math.stackexchange.com/questions/76539. – Qi Zhu Mar 02 '21 at 06:04