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:

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 $M\subset E^{\downarrow }.$

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 (not necessarily unique!) morphism $w$ such that the diagram

commutes.

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

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