↑ arvau

Wikipedia / 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:

  1. E and M both contain all isomorphisms of C and are closed under composition.

  2. Every morphism f of C.

  3. The factorization is functorial making the following diagram commute:

Remark: in the arrow category.

Orthogonality

are said to be orthogonal such that the diagram

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

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

Proof:.

Equivalent definition

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

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:

  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.

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 equivalencess W, fibrations F and cofibrations C so that

  • C has all limitss and colimits,

    is a weak factorization system,

    is a weak factorization system, and

    , then so is the third.

and it is called a trivial cofibration to the terminal object from the initial object is a cofibration.

References