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Factorization system

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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.

Contents

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 for some morphisms e ∈ E and m ∈ M .
  3. The factorization is functorial: if u and v are two morphisms such that v m e = m ′ e ′ u for some morphisms e , e ′ ∈ E and m , m ′ ∈ M , then there exists a unique morphism w making the following diagram commute:


Remark: ( u , v ) is a morphism from m e to m ′ e ′ in the arrow category.

Orthogonality

Two morphisms e and m are said to be orthogonal, denoted e ↓ m , if for every pair of morphisms u and v such that v e = m u 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 ↑ = { e | ∀ h ∈ H , e ↓ h } and H ↓ = { m | ∀ h ∈ H , h ↓ m } .

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

(3') E ⊂ M ↑ and M ⊂ E ↓ .


Proof: In the previous diagram (3), take m := i d ,   e ′ := i d (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 ∘ e with e ∈ E and m ∈ M .
  2. E = M ↑ and M = E ↓ .

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 (resp. 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 wrt the morphisms of M.
  2. The class M is exactly the class of morphisms having the right lifting property wrt the morphisms of E.
  3. Every morphism f of C can be factored as f = m ∘ e for some morphisms e ∈ E and m ∈ M .

References

Factorization system Wikipedia
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