Factorization system

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=m\circ e for some morphisms e\in E and m\in 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: 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 e\downarrow 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=\{e\quad|\quad\forall h\in H, e\downarrow h\} and H^\downarrow=\{m\quad|\quad\forall h\in H, h\downarrow m\}. Since in a factorization system E\cap M contains all the isomorphisms, the condition (3) of the definition is equivalent to :(3') E\subseteq M^\uparrow and M\subseteq E^\downarrow. Proof: In the previous diagram (3), take m:= id ,\ e' := id (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: • Every morphism f of C can be factored as f=m\circ e with e\in E and m\in 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 (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: • The class E is exactly the class of morphisms having the left lifting property with respect to each morphism in M. • The class M is exactly the class of morphisms having the right lifting property with respect to each morphism in E. • Every morphism f of C can be factored as f=m\circ e for some morphisms e\in E and m\in M. 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 • C has all limits and colimits, • (C \cap W, F) is a weak factorization system, • (C, F \cap W) is a weak factorization system, and • W satisfies the two-out-of-three property: if f and g are composable morphisms and two of f,g,g\circ f are in W, then so is the third. 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 X\rightarrow 1 to the terminal object is a fibration, and it is called cofibrant if the morphism 0\rightarrow X from the initial object is a cofibration. == References ==