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Concept# Section (category theory)

Summary

In , a branch of mathematics, a section is a right inverse of some morphism. , a retraction is a left inverse of some morphism.
In other words, if and are morphisms whose composition is the identity morphism on , then is a section of , and is a retraction of .
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).
In algebra, sections are also called split monomorphisms and retractions are also called split epimorphisms. In an , if is a split epimorphism with split monomorphism , then is isomorphic to the direct sum of and the of . The synonym coretraction for section is sometimes seen in the literature, although rarely in recent work.
A section that is also an epimorphism is an isomorphism. Dually a retraction that is also a monomorphism is an isomorphism.
The concept of a retraction in category theory comes from the essentially similar notion of a retraction in topology: where is a subspace of is a retraction in the topological sense, if it's a retraction of the inclusion map 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.
Both use of left/right inverse and section/retraction are commonly seen in the literature: the former use has the advantage that it is familiar from the theory of semigroups and monoids; the latter is considered less confusing by some because one does not have to think about 'which way around' composition goes, an issue that has become greater with the increasing popularity of the synonym f;g for g∘f.

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In , a branch of mathematics, a section is a right inverse of some morphism. , a retraction is a left inverse of some morphism. In other words, if and are morphisms whose composition is the identity morphism on , then is a section of , and is a retraction of . 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). In algebra, sections are also called split monomorphisms and retractions are also called split epimorphisms.

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