Natural Transformation - Definition

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) between objects of 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

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 ηYF(f) = G(f) ∘ 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.

Read more about this topic:  Natural Transformation

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