nLab
split monomorphism

Definition

A split monomorphism in a category $C$ is a morphism $m:A\to B$ which has a retraction, meaning a morphism $r:B\to A$ such that $r\circ m=1_A$.

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

The dual notion is that of split epimorphism.

Properties

Any split monomorphism is automatically a regular monomorphism (it is the equalizer of $m\circ r$ and $1_B$), and therefore also a strong monomorphism, an extremal monomorphism, and (of course) a monomorphism.