 Natural transformation

This article is about natural transformations in category theory. For the natural competence of bacteria to take up foreign DNA, see Transformation (genetics).
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 socalled 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.
Contents
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 with that in D using a morphism η_{X}. η_{X} : F(X) → G(X) called the component of η at X, such that for every morphism f : X → Y 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 : F → G or 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} ∘ F(f) = G(f) ∘ 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 such as
 "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 *^{op} b = 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 *^{op} b) = f(b * a) = f(b) * f(a) = f^{op}(a) *^{op} f^{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} ∘ f = f^{op} ∘ 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 are the components of 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 (necessarily) 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_{X}η_{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 ηε : 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.
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} = Hη_{X}.
If on the other hand K : B → C is a functor, the natural transformation ηK : FK → GK is defined by
Functor categories
Main article: Functor categoryIf 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 forms a category since for any functor F there is an identity natural transformation 1_{F} : F → F (which assigns to every object X the identity morphism on F(X)) and the composition of two natural transformations (the "vertical composition" above) is again a natural transformation.
The isomorphisms in C^{I} are precisely the natural isomorphisms. That is, 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}.
The functor category C^{I} 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 2category Cat whose
 0cells (objects) are the small categories,
 1cells (arrows) between two objects C and D are the functors from C to D,
 2cells between two 1cells (functors) and 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 homcategory in this category (smallness issues aside).
Yoneda lemma
Main article: Yoneda lemmaIf X is an object of a locally small category C, then the assignment Y ↦ 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.
See also
 Extranatural transformation
References
 Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd ed.). SpringerVerlag. ISBN 0387984038.
Categories: Functors
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