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

Orthogonality

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
center
commutes. This notion can be extended to define the orthogonals of sets of morphisms by
Since in a factorization system contains all the isomorphisms, the condition of the definition is equivalent to
Proof: In the previous diagram, take and.

Equivalent definition

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

    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 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.
center
A weak factorization system 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 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.