Section and retraction

Summary

When two morphisms $f \colon A \to B$ and $r \colon B \to A$ satisfy $r \circ f = 1_A$, the morphism $f$ is a section of $r$ and $r$ is a retraction of $f$. A morphism with a retraction is a split monomorphism, and a morphism with a section is a split epimorphism. These are one-sided forms of invertibility, and every functor preserves them. In linear algebra and homological algebra, split monomorphisms and split epimorphisms describe direct summands.