nLab
split epimorphism

Definition

A split epimorphism in a category $C$ is a morphism $e:A\to B$ which has a section, meaning a morphism $s:B\to A$ such that $e\circ s=1_B$.

In such a situation one also says that $B$ is a retract of $A$, and that $B$ is a splitting of the idempotent $s\circ e:A\to A$.

The dual notion is split monomorphism.

Properties

• Any split epimorphism is automatically a regular epimorphism (it is the coequalizer of $s\circ e$ and $1_A$), and therefore also a strong epimorphism, an extremal epimorphism, and (of course) an epimorphism.

• The axiom of choice internal to a category $C$ 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.

Applications

The notion of split epimorphism arises often as a condition on fibrations in categories of chain complexes. See there for details

Examples

• In Vect, every epimorphism is split. For $\varphi :V\to W$ a surjective linear map, we can find an isomorphism $V\simeq \ker (\varphi )\oplus V\prime$. Then $\varphi {\mid }_{V\prime }$ is an isomorphism, and its inverse $W\to V\prime \hookrightarrow \ker (\varphi )\oplus V\prime$ is a section of $\varphi$.