- 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 category theory and consequently appear in the majority of its applications.
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 ηX. ηX : 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 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 ∘ F(f) = G(f) ∘ ηX 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.
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 (Gop,*op) as follows: Gop is the same set as G, and the operation *op is defined by a *op b = b * a. All multiplications in Gop are thus "turned around". Forming the opposite group becomes a (covariant!) functor from Grp to Grp if we define fop = f for any group homomorphism f: G → H. Note that fop is indeed a group homomorphism from Gop to Hop:
- fop(a *op b) = f(b * a) = f(b) * f(a) = fop(a) *op fop(b).
The content of the above statement is:
- "The identity functor IdGrp : Grp → Grp is naturally isomorphic to the opposite functor op : Grp → Grp."
To prove this, we need to provide isomorphisms ηG : G → Gop for every group G, such that the above diagram commutes. Set ηG(a) = a−1. The formulas (ab)−1 = b−1 a−1 and (a−1)−1 = a show that ηG is a group homomorphism which is its own inverse. To prove the naturality, we start with a group homomorphism f : G → H and show ηH ∘ f = fop ∘ ηG, i.e. (f(a))−1 = fop(a−1) for all a in G. This is true since fop = f and every group homomorphism has the property (f(a))−1 = f(a−1).
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 × Abop × Abop → 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 = εXηX. 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 = HηX.
If on the other hand K : B → C is a functor, the natural transformation ηK : FK → GK is defined by
If C is any category and I is a small category, we can form the functor category CI 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 1F : 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 CI 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 = 1G and εη = 1F.
The functor category CI is especially useful if I arises from a directed graph. For instance, if I is the category of the directed graph • → •, then CI has as objects the morphisms of C, and a morphism between φ : U → V and ψ : X → Y in CI 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) 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 CI is then simply a hom-category in this category (smallness issues aside).
If X is an object of a locally small category C, then the assignment Y ↦ HomC(X, Y) defines a covariant functor FX : 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.
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.
- Extranatural transformation
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