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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 : X → Y {\displaystyle f:X\to Y} and g : Y → X {\displaystyle g:Y\to X} are morphisms whose composition f ∘ g : Y → Y {\displaystyle f\circ g:Y\to Y} is the identity morphism on Y {\displaystyle Y} , then g {\displaystyle g} is a section of f {\displaystyle f} , and f {\displaystyle f} is a retraction of g {\displaystyle g}.
Every section is a monomorphism , and every retraction is an epimorphism.
In algebra, sections are also called split monomorphisms and retractions are also called split epimorphisms. In an abelian category, if f : X → Y {\displaystyle f:X\to Y} is a split epimorphism with split monomorphism g : Y → X {\displaystyle g:Y\to X} , then X {\displaystyle X} is isomorphic to the direct sum of Y {\displaystyle Y} and the kernel of f {\displaystyle f}. The synonym coretraction for section is sometimes seen in the literature, although rarely in recent work.