Definitions

# Section (category theory)

In category theory, a branch of mathematics, a section is a right inverse of a morphism. Dually, a retraction is a left inverse. In other words, if $fcolon Xto Y$ and $gcolon Yto X$ are morphisms whose composition $fgcolon Yto Y$ is the identity morphism on Y, then g is a section of f, and f is a retraction of g.

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 $picolon X to bar X$, a section of $pi$ is called a transversal.