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:
#"E" and "M" both contain all isomorphisms of C and are closed under composition.
#Every morphism "f" of C can be factored as $f=mcirc\; e$ for some morphisms $ein\; E$ and $min\; M$.
#The factorization is "functorial": if $u$ and $v$ are two morphisms such that $vme=m\text{'}e\text{'}u$ for some morphisms $e,\; e\text{'}in\; E$ and $m,\; m\text{'}in\; M$, then there exists a unique morphism $w$ making the following diagram commute:

Orthogonality

Two morphisms $e$ and $m$ are said to be "orthogonal", what we write $edownarrow\; m$, 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

:$H^uparrow=\{equad|quadforall\; hin\; H,\; edownarrow\; h\}$ and $H^downarrow=\{mquad|quadforall\; hin\; H,\; hdownarrow\; m\}.$

Since in a factorization system $Ecap\; M$ contains all the isomorphisms, the condition (3) of the definition is equivalent to:(3') $Esubset\; M^uparrow$ and $Msubset\; E^downarrow.$

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:

#Every morphism "f" of C can be factored as $f=mcirc\; e$ with $ein\; E$ and $min\; M.$
#$E=M^uparrow$ 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 :
#The class "E" is exactly the class of morphisms having the left lifting property wrt the morphisms of "M".
#The class "M" is exactly the class of morphisms having the right lifting property wrt the morphisms of "E".
#Every morphism "f" of C can be factored as $f=mcirc\; e$ for some morphisms $ein\; E$ and $min\; M$.

References

* cite journal
author = Peter Freyd, Max Kelly
year = 1972
title = Categories of Continuous Functors I
journal = Journal of Pure and Applied Algebra
volume = 2