Factorization system

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 for some morphisms and .
3. The factorization is functorial: if and are two morphisms such that for some morphisms and , then there exists a unique morphism making the following diagram commute:

Remark: is a morphism from to in the arrow category.

Two morphisms and are said to be orthogonal, denoted , if for every pair of morphisms and such that there is a unique morphism such that the diagram

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

and

Since in a factorization system contains all the isomorphisms, the condition (3) of the definition is equivalent to

(3') and

Proof: In the previous diagram (3), take (identity on the appropriate object) and .

The pair 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 with and
2. and

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:

1. The class E is exactly the class of morphisms having the left lifting property with respect to each morphism in M.
2. The class M is exactly the class of morphisms having the right lifting property with respect to each morphism in E.
3. Every morphism f of C can be factored as for some morphisms and .

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,

• is a weak factorization system,

• is a weak factorization system, and

• satisfies the two-out-of-three property: if and are composable morphisms and two of 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 and it is called a trivial cofibration if it belongs to An object is called fibrant if the morphism to the terminal object is a fibration, and it is called cofibrant if the morphism from the initial object is a cofibration.