A split epimorphism in a category is a morphism which has a section, meaning a morphism such that .
In such a situation one also says that is a retract of , and that is a splitting of the idempotent .
Any split epimorphism is automatically a regular epimorphism (it is the coequalizer of and ), and therefore also a strong epimorphism, an extremal epimorphism, and (of course) an epimorphism.
The axiom of choice internal to a category can be phrased as “all epimorphisms are split.” In Set this is equivalent to the usual axiom of choice; in many other categories it is just false.