Category theory | Homological algebra
Section (category theory)
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. (Wikipedia).
