nLab
retraction

Definition

A retraction of a morphism $f:A\to B$ is a left-inverse: a morphism $\rho :B\to A$ such that

$$A\stackrel{f}{\to }B\stackrel{\rho }{\to }A$$A \stackrel{f}{\to} B \stackrel{\rho}{\to} A

equals the identity morphism on $A$.

In this case, $f$ may also be called a section of $\rho$, $A$ may be called a retract of $B$, and the entire situation is said to split the idempotent

$$B\stackrel{\rho }{\to }A\stackrel{f}{\to }B.$$B \stackrel{\rho}{\to} A \stackrel{f}{\to} B .

A split monomorphism is a morphism that has a retraction; a split epimorphism is a morphism that is a retraction.