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.

A factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that:

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 (3) of the definition is equivalent to

Proof: In the previous diagram (3), take

(identity on the appropriate object) and

of classes of morphisms of C is a factorization system if and only if it satisfies the following conditions: 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] 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 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

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.