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 $f\colon X\to Y$ 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 $g$.

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 section 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, it is also true that 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/2Z which sends every integer to its image modulo 2 does not split; in fact the only morphism Z/2Z→Z is the 0 map. Similarly, the natural monomorphism Z/2Z→Z/4Z doesn't split because there is no non-trivial homomorphism Z/4Z→Z/2Z.

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

See also

• Splitting lemma
• Inverse function#Left and right inverses