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:
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 ∘ e {\displaystyle f=m\circ e} for some morphisms e ∈ E {\displaystyle e\in E} and m ∈ M {\displaystyle m\in M} . 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:
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.}
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:
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