Factorization system

In mathematics, it can be shown that a function can be written as the composite of a surjective function followed by an injective function. Factorization systems are a generalization of this fact in category theory.

Suppose that C is a category. A factorization system (E, M) for C is a pair of classes of morphisms of C such that

• E and M both contain all isomorphisms of C and are closed under composition with those morphisms,
• every morphism f of C can be written as

$f=m\circ e$

with $e\in E$ and $m\in M$, and

• for every morphisms $e:Y\to X\in E$ and $m:X''\to X'\in M$, such that there exists morphisms $v:Y\to X''$ and $f:X\to X'$ satisfying $f\circ e=m\circ v$, there is a unique morphism $h:X\to X''$ such that $h\circ e=v$ and $m\circ h=f$.