A split epimorphism in a category is a morphism in such that there exists a morphism such that the composite equals the identity morphism . Then the morphism , which satisfies the dual condition, is a split monomorphism.
We say that:
is a section of ,
is a retraction of ,
is a retract of ,
Alternatively, it is also possible to define a split epimorphism as an absolute epimorphism: a morphism such that for every functor out of , is an epimorphism. From the definition as a morphism having a section, it is obvious that any split epimorphism is absolute; conversely, that the image of under the representable functor is an epimorphism reduces to the characterization above.
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 may be true without assuming the axiom of choice (in ), or it may be false regardless of the axiom of choice.
The notion of split epimorphism arises often as a condition on fibrations in categories of chain complexes. See there for details.