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: Remark: (u,v) is a morphism from me to m'e' in the arrow category.

Orthogonality

Two morphisms e and m are said to be orthogonal, denoted, 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 Since in a factorization system E\cap M 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 m' := m.

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:

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: 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 F\cap W, and it is called a trivial cofibration if it belongs to C\cap W. An object X 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.