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.

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 diagram

commute.

Orthogonality

Two morphisms $e$ and $m$ are said to be orthogonal, what we write $e\downarrow m$ , 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.