Definition:Section (Category Theory)

Definition

Let $\mathbf C$ be a metacategory.

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

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

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

Also known as

Some authors refer to this as a coretraction.

Also see

• Split Epimorphism, a morphism admitting a section
• Retraction, the name for $f$ in the same situation, from the viewpoint of $g$

Sources

• 1965: Barry Mitchell: Theory of Categories
• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.1.1$: Definition $2.7$