# Definition

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

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

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

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

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

Revised on September 9, 2010 16:52:26 by Toby Bartels (64.89.61.238)