Natural transformation

In category theory, an abstract 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 categorical algebra and consequently appear in the majority of its applications.

Table of contents

1 Definition 2 Examples 3 Yoneda lemma 4 Functor categories 5 Historical Notes

Definition

If F and G are (covariant) 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 η_X : F(X) -> G(X) in D, such that for every morphism f : X -> Y in C we have

η_Y o F(f) = G(f) o η_X.

This equation can conveniently be expressed by the commutative diagram

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

Examples

If K is a field, then for every vector space V over K we have a natural" injective linear map V -> V^** from he vector space into its double dual. These maps are "natural" in the following sense: the double dual operation is a functor, and the maps form a natural transformation from the identity functor to the double dual functor.

Consider the category Ab of abelian groups and group homomorphisms. For all abelian groups X, Y and Z we have a group isomorphism

Hom(X, Hom(Y, Z)) -> Hom(XY, Z).

These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors Ab^op x Ab^op x Ab -> Ab.

Yoneda lemma

If X is an object of the category C, then the assignment Y |-> Mor_C(X, Y) defines a covariant functor F_X : C -> Set. This functor is called representable. The natural transformations from a representable functor to an arbitrary functor F : C -> Set are completely known and easy to describe; this is the content of the Yoneda lemma.

Functor categories

If C is any category and I is a small category, we can form the functor category C^I having as objects all functors from I to C and as morphisms the natural transformations between those functors. This is especially useful if I arises from a directed graph. For instance, if I is the category of the directed graph * -> *, then C^I has as objects the morphisms of C, and a morphism between φ : U -> V and ψ : X -> Y in C^I is a pair of morphisms f : U -> X and g : V -> Y in C such that the "square commutes", i.e. ψ f = g φ.

Historical Notes

Saunders Mac Lane, one of the founders of category theory, is said to have remarked, "I didn't invent categories to study functors; I invented them to study natural transformations." Just as the study of groups is not complete without a study of homomorphisms, so the study of categories is not complete without the study of functors. The reason for Mac Lane's comment is that the study of functors is itself not complete without the study of natural transformations.

The context of Mac Lane's remark was the axiomatic theory of homology. Different ways of constructing homology could be shown to coincide: for example in the case of a simplicial complex the groups defined directly, and those of the singular theory, would be isomorphic. But that in itself stated much less than the existence of a natural transformation of the corresponding homology functors.