Reference articles on history, science, culture and more
Surprise meSurprise meSubscribe
Encyclopedia

Section (category theory)

Right inverse of a morphism

Wikipedia contributorsAdapted for Fathomly · View authors
2 min read
Source updated May 23, 2026
Image credit is listed at the end of this article.

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\to Y and g:Y\to X are morphisms whose composition f\circ g: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 (every morphism with a left inverse is left-cancellative), and every retraction is an epimorphism (every morphism with a right inverse is right-cancellative).

If there exists a section from Y to X, then we say that Y is a retract of X.

In algebra, sections are also called split monomorphisms and retractions are also called split epimorphisms. In an abelian category, if f:X\to Y is a split epimorphism with split monomorphism g:Y\to X, then X is isomorphic to the direct sum of Y and the kernel of f. The synonym coretraction for section is sometimes seen in the literature, although rarely in recent work.

01Properties

02Terminology

The concept of a retraction in category theory comes from the essentially similar notion of a retraction in topology: f:X\to Y where Y is a subspace of X is a retraction in the topological sense, if it's a retraction of the inclusion map i:Y\hookrightarrow X in the category theory sense. The concept in topology was defined by Karol Borsuk in 1931.

Borsuk's student, Samuel Eilenberg, was with Saunders Mac Lane the founder of category theory, and (as the earliest publications on category theory concerned various topological spaces) one might have expected this term to have initially be used. In fact, their earlier publications, up to, e.g., Mac Lane (1963)'s Homology, used the term right inverse. It was not until 1965 when Eilenberg and John Coleman Moore coined the dual term 'coretraction' that Borsuk's term was lifted to category theory in general. The term coretraction gave way to the term section by the end of the 1960s.

03Examples

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/2Z which sends every integer to its remainder modulo 2 does not split; in fact the only morphism Z/2Z → Z is the zero map. Similarly, the natural monomorphism Z/2Z → Z/4Z doesn't split even though there is a non-trivial morphism 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.

Watch videos about Section (category theory)Explainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Section (category theory)”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Continue exploring

Related topics

Splitting lemma

In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence 0 ⟶ A ⟶ q B ⟶ r C ⟶ 0. {\displaystyle 0\longrightarrow A\mathrel {\overset {q}{\longrightarrow }} B\mathrel {\overset {r}{\longrightarrow }} C\longrightarrow 0.} If any of these statements holds, the sequence is called a split exact sequence, and the sequence is said to split.

Inverse function

In mathematics, the inverse function of a function f is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by f − 1 .

Transversal (combinatorics)

In mathematics, particularly in combinatorics, given a family of sets, here called a collection C, a transversal is a set containing exactly one element from each member of the collection. When the sets of the collection are mutually disjoint, each element of the transversal corresponds to exactly one member of C (the set it is a member of).