Factorization system

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Category theory generalization of fumction factorization

title: "Factorization system" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["category-theory"] description: "Category theory generalization of fumction factorization" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Factorization_system" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Category theory generalization of fumction factorization ::

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.

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: ::figure[src="https://upload.wikimedia.org/wikipedia/commons/7/73/Factorization_system_functoriality.png"] ::

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

::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/ac/Factorization_system_orthogonality.png"] ::

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.

::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/ac/Factorization_system_orthogonality.png"] ::

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 | author = Peter Freyd, Max Kelly | year = 1972 | title = Categories of Continuous Functors I | journal = Journal of Pure and Applied Algebra | volume = 2
  • {{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.

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