Factorization system

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:
#"E" and "M" both contain all isomorphisms of C and are closed under composition.
#Every morphism "f" of C can be factored as f=mcirc e for some morphisms ein E and min M.
#The factorization is "functorial": if u and v are two morphisms such that vme=m'e'u for some morphisms e, e'in E and m, m'in M, then there exists a unique morphism w making the following diagram commute:

Orthogonality

Two morphisms e and m are said to be "orthogonal", what we write edownarrow m, 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

:H^uparrow={equad|quadforall hin H, edownarrow h} and H^downarrow={mquad|quadforall hin H, hdownarrow m}.

Since in a factorization system Ecap M contains all the isomorphisms, the condition (3) of the definition is equivalent to:(3') Esubset M^uparrow and Msubset E^downarrow.

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:

#Every morphism "f" of C can be factored as f=mcirc e with ein E and min M.
#E=M^uparrow and M=E^downarrow.

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 :
#The class "E" is exactly the class of morphisms having the left lifting property wrt the morphisms of "M".
#The class "M" is exactly the class of morphisms having the right lifting property wrt the morphisms of "E".
#Every morphism "f" of C can be factored as f=mcirc e for some morphisms ein E and min M.

References

* cite journal
author = Peter Freyd, Max Kelly
year = 1972
title = Categories of Continuous Functors I
journal = Journal of Pure and Applied Algebra
volume = 2


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