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\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:XY is a split epimorphism with split monomorphism g:YX, then X is isomorphic to the direct sum of Y and the kernel of f.

Examples[edit]

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 ZZ/2Z which sends every integer to its image modulo 2 does not split; in fact the only morphism Z/2ZZ is the 0 map. Similarly, the natural monomorphism Z/2ZZ/4Z doesn't split even though there is a non-trivial homomorphism Z/4ZZ/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[edit]