retraction

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)