Factorization system

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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.

Contents 1 Definition 2 Orthogonality 3 Equivalent definition 4 Weak factorization systems 5 References

[edit] 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 can be factored as for some morphisms and .
3. The factorization is functorial: if u and v are two morphisms such that vme = m'e'u for some morphisms and , then there exists a unique morphism w making the following diagram commute:

[edit] Orthogonality

Two morphisms e and m are said to be orthogonal, what we write , if for every pair of morphisms u and v such that ve = mu there is a unique morphism w 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

[edit] 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:

1. Every morphism f of C can be factored as with and
2. and

[edit] 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 (resp. 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 (not necessarily unique!) morphism w such that the diagram

commutes.

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 wrt the morphisms of M.
2. The class M is exactly the class of morphisms having the right lifting property wrt the morphisms of E.
3. Every morphism f of C can be factored as for some morphisms and .

[edit] References

• Peter Freyd, Max Kelly (1972). "Categories of Continuous Functors I". Journal of Pure and Applied Algebra 2.