Epimorphism

Summary

An epimorphism is a morphism $f$ that can be cancelled on the right: $g_1 \circ f = g_2 \circ f$ implies $g_1 = g_2$. It is the categorical version of a surjective function, and the dual notion of a monomorphism. In the categories of sets and of groups, the epimorphisms are exactly the surjective homomorphisms. In other categories an epimorphism need not be surjective: the inclusion of the integers into the rationals is an epimorphism of rings.