Epimorphisms

An epimorphism, or epi, is a morphism f: A → B in a category C that is right-cancellable: for any object X and any pair of morphisms g, h: B → X, if g ∘ f = h ∘ f then g = h.

Equivalently, f is an epimorphism if postcomposition with f does not distinguish maps out of B; that is, f is right-cancelable.

In many familiar categories, epimorphisms coincide with surjective morphisms. For example, in the category Set, a function is an epimorphism precisely when it is surjective. The same holds in the categories of groups, abelian groups, and modules: a homomorphism is an epi if and only if it is surjective.

There are categories in which epis are not surjective on underlying elements. A standard example is the category of rings (with unity) where localization maps are epimorphisms but need not be surjective. For instance, the inclusion Z → Q is an epimorphism in Ring, though it is not surjective as a function.

Dually, a monomorphism is a morphism that is left-cancellable. In abelian categories, epimorphisms can be characterized as morphisms with zero cokernel, and monomorphisms as morphisms with zero kernel. Epimorphisms are closed under composition, and any isomorphism is both an epimorphism and a monomorphism.