# Factorization system

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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 ${\displaystyle f=m\circ e}$ 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.

## Orthogonality

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') ${\displaystyle E\subset M^{\uparrow }}$ and ${\displaystyle M\subset E^{\downarrow }.}$

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

## Equivalent definition

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 ${\displaystyle M=E^{\downarrow }.}$

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

## References

• Peter Freyd, Max Kelly (1972). "Categories of Continuous Functors I". Journal of Pure and Applied Algebra. 2.