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

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 (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 $f=m\circ e$ for some morphisms $e\in E$ and $m\in M$.

References

• Notes, Factorization Systems, Emily Riehl, 2008