Natural transformation

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 categorical algebra and consequently appear in the majority of its applications.

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 called the component of η at X, 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

diagram defining natural transformations

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 : F → G. This is also expressed by saying the family of morphisms η_X : F(X) → G(X) is natural in X.

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 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 η_X: F(X) → G(X). Thus a natural transformation is an infranatural transformation for which η_Y o F(f) = G(f) o η_X for every morphism f : X → Y. 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.

Examples

A worked example

Statements like

"Every group is naturally isomorphic to its opposite group"

abound in modern mathematics. We will now give the precise meaning of this statement as well as its proof. Consider the category Grp of all groups with group homomorphisms as morphisms. If (G,*) is a group, we define its opposite group (G^op,*^op) as follows: G^op is the same set as G, and the operation *^op is defined by a*^opb = b*a. All multiplications in G^op are thus "turned around". Forming the opposite group becomes a (covariant!) functor from Grp to Grp if we define f^op = f for any group homomorphism f: G → H. Note that f^op is indeed a group homomorphism from G^op to H^op:

f^op(a*^opb) = f(b*a) = f(b)*f(a) = f^op(a)*^opf^op(b).

The content of the above statement is:

"The identity functor Id_Grp : Grp → Grp is naturally isomorphic to the opposite functor -^op : Grp → Grp."

To prove this, we need to provide isomorphisms η_G : G → G^op for every group G, such that the above diagram commutes. Set η_G(a) = a^-1. The formulas (ab)^-1 = b^-1 a^-1 and (a^-1)^-1 = a show that η_G is a group homomorphism which is its own inverse. To prove the naturality, we start with a group homomorphism f : G → H and show η_H o f = f^op o η_G, i.e. (f(a))^-1 = f^op(a^-1) for all a in G. This is true since f^op = f and every group homomorphism has the property (f(a))^-1 = f(a^-1).

Further examples

If K is a field, then for every vector space V over K we have a "natural" injective linear map V → V** from the 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.

Every finite dimensional vector space is also isomorphic to its dual space. But this isomorphism relies on an arbitrary choice of basis, and is not natural, though there is an infranatural transformation. More generally, any vector spaces with the same dimensionality are isomorphic, but not naturally so. (Note however that if the space has a nondegenerate bilinear form, then there is a natural isomorphism between the space and its dual. Here the space is viewed as an object in the category of vector spaces and transposes of maps.)

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⊗Y, Z) → Hom(X, Hom(Y, Z)).

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

Natural transformations arise frequently in conjunction with adjoint functors. Indeed, adjoint functors are defined by a certain natural isomorphism. Additionally, every pair of adjoint functors comes equipped with two natural transformations (generally not isomorphisms) called the unit and counit.

Operations with natural transformations

If η : F → G and ε : G → H are natural transformations between functors F,G,H:C → D, then we can compose them to get a natural transformation εη : F → H. This is done componentwise: (εη)_X = ε_Xη_X. This "vertical composition" of natural transformation is associative and has an identity, and allows one to consider the collection of all functors C → D itself as a category (see below under Functor categories).

Natural transformations also have a "horizontal composition". If η:F → G is a natural transformation between functors F,G:C → D and ε: J → K is a natural transformation between functors J,K:D → E, then the composition of functors allows a composition of natural transformations η ˆ ε: JF → KG. This operation is also associative with identity, and the identity coincides with that for vertical composition. The two operations are related by an identity which exchanges vertical composition with horizontal composition.

A natural transformation η : F → G is a natural isomorphism if and only if there exists a natural transformation ε : G → F such that ηε = 1_G and εη = 1_F (where 1_F : F → F is the natural transformation assigning to every object X the identity morphism on F(X)).

If η : F → G is a natural transformation between functors F,G:C → D, and H : D → E is another functor, then we can form the natural transformation Hη : HF → HG by defining (Hη)_X = H(η_X). If on the other hand K : B → C is a functor, the natural transformation ηK : FK → GK is defined by (ηK)_X = η_K(X).

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.

More generally, one can build the 2-category Cat whose

0-cells (objects) are the small categories,
1-cells (arrows) between two objects $C$ and $D$ are the functors from $C$ to $D$ ,
2-cells between two 1-cells (functors) $F:C\to D$ and $G:C\to D$ are the natural transformations from $F$ to $G$ .

The horizontal and vertical compositions are the compositions between natural transformations described previously. A functor category $C^{I}$ is then simply a hom-set in this category (smallness issues aside).

Yoneda lemma

If X is an object of the category C, then the assignment Y $\mapsto$ Hom_C(X, Y) defines a covariant functor F_X : C → Set. This functor is called representable (more generally, a representable functor is any functor naturally isomorphic to this functor for an appropriate choice of X). 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.

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. What cannot easily be expressed without the language of natural transformations is how homology groups are compatible with morphisms between objects, and how two equivalent homology theories not only have the same homology groups, but also the same morphisms between those groups.