Section (category theory)

In the mathematical field of category theory, given a pair of maps $f\colon X\to Y$, $g\colon Y\to X$ such that $fg=\operatorname {id} _{Y}$ (the identity map on Y), we call g a section of f, and f a retraction of g. In other words, a section is a right inverse, and a retraction is a left inverse (and these are dual notions).

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.

A section is a monomorphism, and a 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 $\pi$ is called a '''transversal'''.

See also

• Splitting lemma