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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.
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 ) {\displaystyle (u,v)} is a morphism from m e {\displaystyle me} to m ′ e ′ {\displaystyle m'e'} in the arrow category.
Two morphisms e {\displaystyle e} and m {\displaystyle m} are said to be orthogonal, denoted e ↓ m {\displaystyle e\downarrow m} , if for every pair of morphisms u {\displaystyle u} and v {\displaystyle v} such that v e = m u {\displaystyle ve=mu} there is a unique morphism w {\displaystyle 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 ∩ M {\displaystyle E\cap M} contains all the isomorphisms, the condition (3) of the definition is equivalent to
Proof: In the previous diagram (3), take m := i d , e ′ := i d {\displaystyle m:=id,\ e':=id} (identity on the appropriate object) and m ′ := m {\displaystyle m':=m} .
The pair ( E , M ) {\displaystyle (E,M)} 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 F ∩ W , {\displaystyle F\cap W,} and it is called a trivial cofibration if it belongs to C ∩ W . {\displaystyle C\cap W.} An object X {\displaystyle X} is called fibrant if the morphism X → 1 {\displaystyle X\rightarrow 1} to the terminal object is a fibration, and it is called cofibrant if the morphism 0 → X {\displaystyle 0\rightarrow X} from the initial object is a cofibration.[3]
