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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 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).

Attributes	Values
rdfs:label	Section (théorie des catégories) (fr) Section (category theory) (en)
rdfs:comment	Dans le domaine mathématique de la théorie des catégories, si on a un couple de morphismes , tel que (le morphisme identité de Y, souvent réalisé par l'application identité sur Y),on dit que g est une section de f, et que f est une rétraction de g. En d'autres termes, une section est un inverse à droite, et une rétraction est un inverse à gauche (ce sont deux notions duales). Le concept au sens des catégories de ces notions est particulièrement important en algèbre homologique, et est étroitement lié à la notion de section d'un fibré en topologie. (fr) 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 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). (en)
foaf:depiction
dcterms:subject	Category theory Homological algebra
Wikipage page ID	10630303 (xsd:integer)
Wikipage revision ID	1093724697 (xsd:integer)
Link from a Wikipage to another Wikipage	Samuel Eilenberg Saunders Mac Lane Epimorphism Monomorphism Basis (linear algebra) Mathematics Function (mathematics) Modular arithmetic Monoid Morphism Empty set Category of sets Topology Domain of a function Dual (category theory) Linear map Semigroup Algebra Fiber bundle Field (mathematics) Direct sum Isomorphism Quotient space (topology) Retraction (topology) Inverse function Isomorphic Zero morphism Category theory Homological algebra Abelian category Academic Press John Moore Karol Borsuk Kernel (category theory) Surjective function Homological algebra Transversal (combinatorics) Axiom of choice Splitting lemma Identity morphism Integer Category of abelian groups Category theory Section (fiber bundle) Category of vector spaces Injective Right-cancellative Springer Verlag Left-cancellative
sameAs	Section (category theory) Section (category theory) Section (category theory) Section (category theory) Section (category theory) Section (category theory)
dbp:wikiPageUsesTemplate	dbt:Cite_book dbt:Reflist
thumbnail	wiki-commons:Special:FilePath/Section_retract.svg?width=300
has abstract	Dans le domaine mathématique de la théorie des catégories, si on a un couple de morphismes , tel que (le morphisme identité de Y, souvent réalisé par l'application identité sur Y),on dit que g est une section de f, et que f est une rétraction de g. En d'autres termes, une section est un inverse à droite, et une rétraction est un inverse à gauche (ce sont deux notions duales). Le concept au sens des catégories de ces notions est particulièrement important en algèbre homologique, et est étroitement lié à la notion de section d'un fibré en topologie. Toute section est un monomorphisme et toute rétraction est un épimorphisme. Elles sont respectivement appelées split mono et split epi. Même dans le cas de la catégorie des ensembles, il n'y a nullement unicité, par exemple, si f est une surjection mais pas une bijection, on peut construire (en admettant l'axiome du choix) plusieurs sections de f. (fr) 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 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 abelian category, if is a split epimorphism with split monomorphism , then is isomorphic to the direct sum of and the kernel of . The synonym coretraction for section is sometimes seen in the literature, although rarely in recent work. (en)
gold:hypernym	Inverse
prov:wasDerivedFrom	wikipedia-en:Section_(category_theory)?oldid=1093724697&ns=0
page length (characters) of wiki page	5912 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Section_(category_theory)
is Link from a Wikipage to another Wikipage of	Epimorphism Monomorphism Retraction (category theory) Ring extension Characteristic class Invariant differential operator Profinite group Transversality (mathematics) Normal bundle Glossary of category theory Morphism Equivalence class Comma category Federer–Morse theorem Coretraction First class constraint Foliation Gerbe Gluing axiom Projective object Rational point Hilbert's Nullstellensatz Inverse element Inverse function Right inverse Chern class

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