Jump to content

Factorization system

From Wikipedia, the free encyclopedia
(Redirected from Weak 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

[edit]

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.

Orthogonality

[edit]

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 .

Equivalent definition

[edit]

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

[edit]

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]

  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.[2]

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.[3]

References

[edit]
  1. ^ Riehl (2014, §11.2)
  2. ^ Riehl (2014, §11.3)
  3. ^ Valery Isaev - On fibrant objects in model categories.
  • Peter Freyd, Max Kelly (1972). "Categories of Continuous Functors I". Journal of Pure and Applied Algebra. 2.
  • Riehl, Emily (2014), Categorical homotopy theory, Cambridge University Press, doi:10.1017/CBO9781107261457, ISBN 978-1-107-04845-4, MR 3221774
[edit]