Section (category theory)


Section (category theory)

In category theory, a branch of mathematics, a section (or coretraction) is a right inverse of a morphism. Dually, a retraction (or retract) is a left inverse. In other words, if f\colon X\to Y and g\colon Y\to X are morphisms whose composition f\circ g\colon Y\to Y is the identity morphism on Y, then g is a section of f, and f is a retraction of g.

If section of a morphism exists, it is called sectionable. Dually, if retraction of a morphism exists, it is called retractable.

The categorical concept of a section is important in homological algebra, and is also closely related to the notion of a section of a fiber bundle in topology: in the latter case, a section of a fiber bundle is a section of the bundle projection map of the fiber bundle.

Every section is a monomorphism, and every retraction is an epimorphism; they are called respectively a split monomorphism and a split epimorphism (the inverse is the splitting).

Examples

Given a quotient space \bar X with quotient map \pi\colon X \to \bar X, a section of π is called a transversal.

See also


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