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.

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
Image:Factorization_system_functoriality.png
commute.

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[edit] 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

Image:Factorization_system_orthogonality.png

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.

[edit] 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.

The condition (2) above can be shown to be equivalent to the condition

(2') E\cap M is exactly the class of isomorphisms of C.

[edit] 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

Image:Factorization_system_orthogonality.png

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.

[edit] References

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