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 ${\displaystyle f:X\to Y}$ and ${\displaystyle g:Y\to X}$ are morphisms whose composition ${\displaystyle f\circ g:Y\to Y}$ is the identity morphism on ${\displaystyle Y}$, then ${\displaystyle g}$ is a section of ${\displaystyle f}$, and ${\displaystyle f}$ is a retraction of ${\displaystyle g}$.^[1]

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 ${\displaystyle f:X\to Y}$ is a split epimorphism with split monomorphism ${\displaystyle g:Y\to X}$, then ${\displaystyle X}$ is isomorphic to the direct sum of ${\displaystyle Y}$ and the kernel of ${\displaystyle f}$. The synonym coretraction for section is sometimes seen in the literature, although rarely in recent work.