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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. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ...
In category theory, a functor is a special type of mapping between categories. ...
In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...
In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ...
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 ηX : F(X) → G(X) in D, such that for every morphism f : X → Y in C we have ηY O F(f) = G(f) O ηX. This equation can conveniently be expressed by the commutative diagram For functors in computer science, see the function object article. ...
In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...
In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...
 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 ηX : F(X) → G(X) is natural in X. Diagram illustrating the defining property of natural transformations, produced with the TeX package XYpic using the following source code. ...
For functors in computer science, see the function object article. ...
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. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...
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 (Gop,*op) as follows: Gop is the same set as G, and the operation *op is defined by a*opb = 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: In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
- fop(a*opb) = f(b*a) = f(b)*f(a) = fop(a)*opfop(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 o f = fop o 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).
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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...
Consider the category Ab of abelian groups and group homomorphisms. For all abelian groups X, Y and Z we have a group isomorphism In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...
- Hom(X, Hom(Y, Z)) → Hom(X⊗Y, Z).
These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors Abop × Abop × Ab → 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 come equipped with two natural transformations (generally not isomorphisms) called the unit and counit. The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...
Operations with natural transformations If η : F → G and ε : G → H are natural transformations between functors C → D, then we can compose them to get a natural transformation εη : F → H. This is done componentwise: (εη)X = εXηX. This composition of natural transformation is associative, and allows to consider the collection of all functors C → D itself as a category (see below under Functor categories). In mathematics, associativity is a property that a binary operation can have. ...
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 (where 1F : 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 C → D, and H : D → E is another functor, then we can form the natural transformation Hη : HF → HG by defining (Hη)X = H(ηX). If on the other hand K : B → C is a functor, the natural transformation ηK : FK → GK is defined by (ηK)X = ηK(X).
Functor categories 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 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. In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ...
This article just presents the basic definitions. ...
Yoneda lemma If X is an object of the category C, then the assignment Y |-> MorC(X, Y) defines a covariant functor FX : 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. In category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. ...
In mathematics, the Yoneda lemma in category theory is an abstract result on functors of the type morphisms into a fixed object. ...
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. Saunders Mac Lane (4 August 1909 - 14 April 2005) was a US mathematician. ...
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
In category theory, a functor is a special type of mapping between categories. ...
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. In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...
In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ...
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