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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 ∘ e {\displaystyle f=m\circ e} for some morphisms e ∈ E {\displaystyle e\in E} and m ∈ M {\displaystyle m\in M} .
3. The factorization is functorial: if u {\displaystyle u} and v {\displaystyle v} are two morphisms such that v m e = m ′ e ′ u {\displaystyle vme=m'e'u} for some morphisms e , e ′ ∈ E {\displaystyle e,e'\in E} and m , m ′ ∈ M {\displaystyle m,m'\in M} , then there exists a unique morphism w {\displaystyle w} making the following diagram commute:

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

Orthogonality

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

H ↑ = { e | ∀ h ∈ H , e ↓ h } {\displaystyle H^{\uparrow }=\{e\quad |\quad \forall h\in H,e\downarrow h\}} and H ↓ = { m | ∀ h ∈ H , h ↓ m } . {\displaystyle H^{\downarrow }=\{m\quad |\quad \forall h\in H,h\downarrow m\}.}

Since in a factorization system E ∩ M {\displaystyle E\cap M} contains all the isomorphisms, the condition (3) of the definition is equivalent to

(3') E ⊆ M ↑ {\displaystyle E\subseteq M^{\uparrow }} and M ⊆ E ↓ . {\displaystyle M\subseteq E^{\downarrow }.}

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} .

Equivalent definition

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:

1. Every morphism f of C can be factored as f = m ∘ e {\displaystyle f=m\circ e} with e ∈ E {\displaystyle e\in E} and m ∈ M . {\displaystyle m\in M.}
2. E = M ↑ {\displaystyle E=M^{\uparrow }} and M = E ↓ . {\displaystyle 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

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 ∘ e {\displaystyle f=m\circ e} for some morphisms e ∈ E {\displaystyle e\in E} and m ∈ M {\displaystyle 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 ∩ W , F ) {\displaystyle (C\cap W,F)} is a weak factorization system,

• ( C , F ∩ W ) {\displaystyle (C,F\cap W)} is a weak factorization system, and

• W {\displaystyle W} satisfies the two-out-of-three property: if f {\displaystyle f} and g {\displaystyle g} are composable morphisms and two of f , g , g ∘ f {\displaystyle f,g,g\circ f} are in W {\displaystyle W} , then so is the third.²

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

• Peter Freyd, Max Kelly (1972). "Categories of Continuous Functors I". Journal of Pure and Applied Algebra. 2.
• Riehl, Emily (2014), Categorical homotopy theory, Cambridge University Press, doi:10.1017/CBO9781107261457, ISBN 978-1-107-04845-4, MR 3221774

External links

• Riehl, Emily (2008), Factorization Systems (PDF)

References

1. Riehl (2014, §11.2) - Riehl, Emily (2014), Categorical homotopy theory, Cambridge University Press, doi:10.1017/CBO9781107261457, ISBN 978-1-107-04845-4, MR 3221774 https://doi.org/10.1017%2FCBO9781107261457 ↩

2. Riehl (2014, §11.3) - Riehl, Emily (2014), Categorical homotopy theory, Cambridge University Press, doi:10.1017/CBO9781107261457, ISBN 978-1-107-04845-4, MR 3221774 https://doi.org/10.1017%2FCBO9781107261457 ↩

3. Valery Isaev - On fibrant objects in model categories. ↩