Factorsuch

FactorSuch is a term used in abstract mathematics to describe a canonical factorization of a morphism through an intermediate object, characterized by a universal property. In a category C, a morphism h: X -> Y is said to have a FactorSuch factorization if there exists an object Z and morphisms f: X -> Z and g: Z -> Y such that h = g ∘ f, and such that for any other factorization h = g' ∘ f' with f': X -> W and g': W -> Y, there exists a unique morphism u: Z -> W with f' = u ∘ f and g' = g ∘ u. Thus Z is initial among all factorizations of h.

Relation to other notions: FactorSuch is closely related to universal properties and to factorization systems. It formalizes the idea of a most efficient or canonical intermediate stage through which a map factors.

Examples: In Set, every function factors as X --p--> im(h) --i--> Y, with p surjective and i injective; the factorization through the image is initial among all factorizations, making it a FactorSuch factorization. In the category of groups, a homomorphism f: G -> H factors through G/ker(f) via the natural quotient map followed by the induced map to H; this also exhibits a universal factoring pattern.

Limitations and usage: Not every morphism has such a universal factorization in all categories. FactorSuch is mainly a useful framework for discussing canonical factorizations and their implications in category theory and algebra.

History: The term FactorSuch appears in contemporary expository writing as a pedagogical label for universal factorization patterns.