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:
"Remark:" formula_10 is a morphism from formula_11 to formula_12 in the arrow category.
Orthogonality.
Two morphisms formula_13 and formula_14 are said to be "orthogonal", denoted formula_15, if for every pair of morphisms formula_4 and formula_5 such that formula_18 there is a unique morphism formula_9 such that the diagram
commutes. This notion can be extended to define the orthogonals of sets of morphisms by
Since in a factorization system formula_22 contains all the isomorphisms, the condition (3) of the definition is equivalent to
"Proof:" In the previous diagram (3), take formula_25 (identity on the appropriate object) and formula_26.
Equivalent definition.
The pair formula_27 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 formula_38 and it is called a trivial cofibration if it belongs to formula_39 An object formula_40 is called fibrant and the morphism formula_41 to the terminal object is a fibration, and it is called cobrant if the morphism formula_42 from the initial object is a cofibration.