Factorization system

Definition

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

1. E and M both contain all isomorphisms of C and are closed under composition.
2. Every morphism f of C can be factored as f=m\circ e for some morphisms e\in E and m\in M.
3. 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:

1. Every morphism f of C can be factored as f=m\circ e with e\in E and m\in M.
2. 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:

1. The class E is exactly the class of morphisms having the left lifting property with respect to each morphism in M.
2. The class M is exactly the class of morphisms having the right lifting property with respect to each morphism in E.
3. 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

• {{cite journal
• {{Citation| author = Riehl|first=Emily|authorlink = Emily Riehl| title = Categorical homotopy theory | publisher = Cambridge University Press| year = 2014| isbn = 978-1-107-04845-4| mr = 3221774| doi = 10.1017/CBO9781107261457}}

References

1. {{harvtxt. Riehl. 2014
2. {{harvtxt. Riehl. 2014
3. Valery Isaev - On fibrant objects in model categories.