Factorization system

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.

Definition
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 $$f=m\circ e$$ for some morphisms $$e\in E$$ and $$m\in M$$.
 * 3) The factorization is functorial: if $$u$$ and $$v$$ are two morphisms such that $$vme=m'e'u$$ for some morphisms $$e, e'\in E$$ and $$m, m'\in M$$, then there exists a unique morphism $$w$$ making the following diagram commute:

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

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



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


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

Since in a factorization system $$E\cap M$$ contains all the isomorphisms, the condition (3) of the definition is equivalent to
 * (3') $$E\subseteq M^\uparrow$$ and $$M\subseteq E^\downarrow.$$

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

Equivalent definition
The pair $$(E,M)$$ 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 $$f=m\circ e$$ with $$e\in E$$ and $$m\in M.$$
 * 2) $$E=M^\uparrow$$ and $$M=E^\downarrow.$$

Weak factorization systems
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: 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
 * 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 $$f=m\circ e$$ for some morphisms $$e\in E$$ and $$m\in M$$.


 * C has all limits and colimits,


 * $$(C \cap W, F)$$ is a weak factorization system,


 * $$(C, F \cap W)$$ is a weak factorization system, and


 * $$W$$ satisfies the two-out-of-three property: if $$f$$ and $$g$$ are composable morphisms and two of $$f,g,g\circ f$$ are in $$W$$, then so is the third.

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 $$F\cap W,$$ and it is called a trivial cofibration if it belongs to $$C\cap W.$$ An object $$X$$ is called fibrant if the morphism $$X\rightarrow 1$$ to the terminal object is a fibration, and it is called cofibrant if the morphism $$0\rightarrow X$$ from the initial object is a cofibration.