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 fact 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 written as

f=m\circ e

for some morphisms

e\in E

and

m\in M

.

(3) 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

TODO

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 $vm=eu$ there is a unique morphism $w$ such that the diagram