Natural transformation

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In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e. the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most basic notions of category theory and consequently appear in the majority of its applications.

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Definition

If F and G are functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism ηX : F(X) → G(X) in D called the component of η at X, such that for every morphism f : XY in C we have:

This equation can conveniently be expressed by the commutative diagram

Natural transformation.svg

If both F and G are contravariant, the horizontal arrows in this diagram are reversed. If η is a natural transformation from F to G, we also write η : FG or η : FG. This is also expressed by saying the family of morphisms ηX : F(X) → G(X) is natural in X.

If, for every object X in C, the morphism ηX is an isomorphism in D, then η is said to be a natural isomorphism (or sometimes natural equivalence or isomorphism of functors). Two functors F and G are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from F to G.

An infranatural transformation η from F to G is simply a family of morphisms ηX: F(X) → G(X). Thus a natural transformation is an infranatural transformation for which ηY o F(f) = G(f) o ηX for every morphism f : XY. The naturalizer of η, nat(η), is the largest subcategory of C containing all the objects of C on which η restricts to a natural transformation.

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