# Section (category theory)

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In category theory, a branch of mathematics, a **section** is a right inverse of some morphism. Dually, a **retraction** is a left inverse of some morphism.
In other words, if *f* : *X* → *Y* and *g* : *Y* → *X* are morphisms whose composition *f* o *g* : *Y* → *Y* is the identity morphism on *Y*, then *g* is a section of *f*, and *f* is a retraction of *g*.

Every section is a monomorphism, and every retraction is an epimorphism.

In algebra the sections are also called **split monomorphisms** and the retractions **split epimorphisms**.
In an abelian category, if *f* : *X* → *Y* is a split epimorphism with split monomorphism *g* : *Y* → *X*,
then *X* is isomorphic to the direct sum of *Y* and the kernel of *f*.

## Examples

In the category of sets, every monomorphism (injective function) with a non-empty domain is a section and every epimorphism (surjective function) is a retraction; the latter statement is equivalent to the axiom of choice.

In the category of vector spaces over a field *K*, every monomorphism and every epimorphism splits; this follows from the fact that linear maps can be uniquely defined by specifying their values on a basis.

In the category of abelian groups, the epimorphism **Z**→**Z**/2**Z** which sends every integer to its image modulo 2 does not split; in fact the only morphism **Z**/2**Z**→**Z** is the 0 map. Similarly, the natural monomorphism **Z**/2**Z**→**Z**/4**Z** doesn't split even though there is a non-trivial homomorphism **Z**/4**Z**→**Z**/2**Z**.

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.

Given a quotient space with quotient map , a section of is called a transversal.

## See also

de:Retraktion und Koretraktion fr:Section (théorie des catégories) pl:Retrakcja (teoria kategorii)