Factorization system explained

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=m\circe

for some morphisms

e\inE

and

m\inM

The factorization is functorial: if

and

are two morphisms such that

vme=m'e'u

for some morphisms

e,e'\inE

and

m,m'\inM

, then there exists a unique morphism

making the following diagram commute:

Remark:

(u,v)

is a morphism from

m'e'

in the arrow category.

Orthogonality

Two morphisms

and

are said to be orthogonal, denoted

e\downarrowm

, if for every pair of morphisms

and

such that

ve=mu

there is a unique morphism

such that the diagram

commutes. This notion can be extended to define the orthogonals of sets of morphisms by

H^{\uparrow=\{e | \forall}h\inH,e\downarrowh\}

and

H^{\downarrow=\{m | \forall}h\inH,h\downarrowm\}.

Since in a factorization system

E\capM

contains all the isomorphisms, the condition (3) of the definition is equivalent to

(3')

E\subseteqM^\uparrow

and

M\subseteqE^\downarrow.

Proof: In the previous diagram (3), take

m:=id, e':=id

(identity on the appropriate object) and

m':=m

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=m\circe

with

e\inE

and

m\inM.

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 (respectively 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 morphism w such that the following diagram commutes. The difference with orthogonality is that w is not necessarily unique.

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 with respect to each morphism in M.
The class M is exactly the class of morphisms having the right lifting property with respect to each morphism in E.
Every morphism f of C can be factored as

f=m\circe

for some morphisms

e\inE

and

m\inM

.This notion leads to a succinct definition of model categories: a model category is a pair consisting of a category C and classes of (so-called) weak equivalences W, fibrations F and cofibrations C so that

C has all limits and colimits,

(C\capW,F)

is a weak factorization system,

(C,F\capW)

is a weak factorization system, and

satisfies the two-out-of-three property: if

and

are composable morphisms and two of

f,g,g\circf

are in

, then so is the third.

A model category is a complete and cocomplete category equipped with a model structure. A map is called a trivial fibration if it belongs to

F\capW,

and it is called a trivial cofibration if it belongs to

C\capW.

An object

is called fibrant if the morphism

X → 1

to the terminal object is a fibration, and it is called cofibrant if the morphism

0 → X

from the initial object is a cofibration.^[1]

References

Valery Isaev - On fibrant objects in model categories.

. 1972 . Categories of Continuous Functors I . Journal of Pure and Applied Algebra . 2 .