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

A factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that:

1. E and M both contain all isomorphisms of C and are closed under composition.
2. Every morphism f of C can be factored as ${\displaystyle f=m\circ e}$ for some morphisms ${\displaystyle e\in E}$ and ${\displaystyle m\in M}$.
3. The factorization is functorial: if ${\displaystyle u}$ and ${\displaystyle v}$ are two morphisms such that ${\displaystyle vme=m'e'u}$ for some morphisms ${\displaystyle e,e'\in E}$ and ${\displaystyle m,m'\in M}$, then there exists a unique morphism ${\displaystyle w}$ making the following diagram commute:

Remark: ${\displaystyle (u,v)}$ is a morphism from ${\displaystyle me}$ to ${\displaystyle m'e'}$ in the arrow category.

Two morphisms ${\displaystyle e}$ and ${\displaystyle m}$ are said to be orthogonal, denoted ${\displaystyle e\downarrow m}$, if for every pair of morphisms ${\displaystyle u}$ and ${\displaystyle v}$ such that ${\displaystyle ve=mu}$ there is a unique morphism ${\displaystyle w}$ such that the diagram

commutes. This notion can be extended to define the orthogonals of sets of morphisms by

${\displaystyle H^{\uparrow }=\{e\quad |\quad \forall h\in H,e\downarrow h\}}$ and ${\displaystyle H^{\downarrow }=\{m\quad |\quad \forall h\in H,h\downarrow m\}.}$

Since in a factorization system ${\displaystyle E\cap M}$ contains all the isomorphisms, the condition (3) of the definition is equivalent to

(3') ${\displaystyle E\subseteq M^{\uparrow }}$ and ${\displaystyle M\subseteq E^{\downarrow }.}$

Proof: In the previous diagram (3), take ${\displaystyle m:=id,\ e':=id}$ (identity on the appropriate object) and ${\displaystyle m':=m}$.

The pair ${\displaystyle (E,M)}$ of classes of morphisms of C is a factorization system if and only if it satisfies the following conditions:

1. Every morphism f of C can be factored as ${\displaystyle f=m\circ e}$ with ${\displaystyle e\in E}$ and ${\displaystyle m\in M.}$
2. ${\displaystyle E=M^{\uparrow }}$ and ${\displaystyle M=E^{\downarrow }.}$

Suppose e and m are two morphisms in a category C. Then e has the left lifting property with respect to m (respectively m has the right lifting property with respect to e) when for every pair of morphisms u and v such that ve = mu there is a morphism w such that the following diagram commutes. The difference with orthogonality is that w is not necessarily unique.

A weak factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that:^[1]

1. The class E is exactly the class of morphisms having the left lifting property with respect to each morphism in M.
2. The class M is exactly the class of morphisms having the right lifting property with respect to each morphism in E.
3. Every morphism f of C can be factored as ${\displaystyle f=m\circ e}$ for some morphisms ${\displaystyle e\in E}$ and ${\displaystyle m\in M}$.

This notion leads to a succinct definition of model categories: a model category is a pair consisting of a category C and classes of (so-called) weak equivalences W, fibrations F and cofibrations C so that

• C has all limits and colimits,

• ${\displaystyle (C\cap W,F)}$ is a weak factorization system,

• ${\displaystyle (C,F\cap W)}$ is a weak factorization system, and

• ${\displaystyle W}$ satisfies the two-out-of-three property: if ${\displaystyle f}$ and ${\displaystyle g}$ are composable morphisms and two of ${\displaystyle f,g,g\circ f}$ are in ${\displaystyle W}$, then so is the third.^[2]

A model category is a complete and cocomplete category equipped with a model structure. A map is called a trivial fibration if it belongs to ${\displaystyle F\cap W,}$ and it is called a trivial cofibration if it belongs to ${\displaystyle C\cap W.}$ An object ${\displaystyle X}$ is called fibrant if the morphism ${\displaystyle X\rightarrow 1}$ to the terminal object is a fibration, and it is called cofibrant if the morphism ${\displaystyle 0\rightarrow X}$ from the initial object is a cofibration.^[3]

• Riehl, Emily (2008), Factorization Systems (PDF)