nLab
retraction

Definition

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

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

equals the identity morphism on AA.

In this case, ff may also be called a section of ρ\rho, AA may be called a retract of BB, and the entire situation is said to split the idempotent

BρAfB. 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)