Definition:Retraction

Definition

Let $\mathbf C$ be a metacategory.

Let $f: C \to D$ be a morphism of $\mathbf C$.

A retraction of $f$ is a morphism $g: D \to C$ such that:

$g \circ f = \operatorname{id}_C$

Retract

Let $g: D \to C$ be a retraction of $f$.

Then $D$ is said to be a retract of $C$.

Also see

• Split Monomorphism, a morphism admitting a retraction
• Section (Category Theory), the name for $f$ in the same situation, from the viewpoint of $g$

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.1.1$: Definition $2.7$