Factorization system

Summary

A factorization system on a category is a pair $(E, M)$ of classes of morphisms. Every morphism factors as a member of $E$ followed by a member of $M$, in an essentially unique way. The two classes determine each other through a unique lifting property. The standard example in sets is (surjections, injections), which factors a function through its image. A weaker variant, the weak factorization system, drops uniqueness and is a basic ingredient of model categories.