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. 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 for some morphisms and . The factorization is functorial: if and are two morphisms such that for some morphisms and , then there exists a unique morphism making the following diagram commute: Remark: is a morphism from to in the arrow category. Orthogonality Two morphisms and are said to be orthogonal, denoted , if for every pair of morphisms and such that there is a unique morphism such that the diagram commutes. This notion can be extended to define the orthogonals of sets of morphisms by and Since in a factorization system contains all the isomorphisms, the condition (3) of the definition is equivalent to (3') and Proof: In the previous diagram (3), take (identity on the appropriate object)