In mathematics, the Yoneda lemma in category theory is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory (a group being a category with just one object). It allows the embedding of any category into a category of functors defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In group theory, Cayleys theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G. This can be understood as an example of the group action of G on the elements of G. A permutation of a... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... 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. ... In mathematics, especially in category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. ... 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. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... Yoneda Nobuo (米田 信夫, March 28, 1930 – April 21, 1996) was a Japanese mathematician and computer scientist. ...

Generalities

The Yoneda lemma suggests that instead of studying the (small) category C, one should study the category of all functors of C into Set (the category of sets with functions as morphisms). Set is a category we understand well, and a functor of C into Set can be seen as a "representation" of C in terms of known structures. The original category C is contained in this functor category, but new objects appear in the functor category which were absent and "hidden" in C. Treating these new objects just like the old ones often unifies and simplifies the theory. In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

This approach is akin to (and in fact generalizes) the common method of studying a ring by investigating the modules over that ring. The ring takes the place of the category C, and the category of modules over the ring is a category of functors defined on C. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ...

Formal statement

General version

Yoneda's lemma concerns functors from a fixed category C to the category of sets, Set. If C is a locally small category (i.e. the hom-sets are actual sets and not proper classes), then each object A of C induces a natural functor to Set called a hom-functor. This functor is denoted: In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... This is a glossary of properties and concepts in category theory in mathematics. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... In mathematics, specifically in category theory, Hom-sets, i. ...

The hom-functor h_A sends X to the set of morphisms Hom(A,X). In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

Let F be an arbitrary functor from C to Set. Then Yoneda's lemma says that for each object A of C the natural transformations from h_A to F are in one-to-one correspondence with the elements of F(A). That is, 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. ...

Given a natural transformation Φ from h_A to F, the corresponding element of F(A) is u = Φ_A(id_A).

There is a contravariant version of Yoneda's lemma which concerns contravariant functors from C to Set. This version involves the contravariant hom-functor For functors in computer science, see the function object article. ...

which sends X to the hom-set Hom(X,A). Given an arbitrary contravariant functor G from C to Set, Yoneda's lemma asserts that

Proof

The proof of Yoneda's lemma is indicated by the following commutative diagram: 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. ...

This diagram shows that the natural transformation Φ is completely determined by u = Φ_A(id_A) since for each morphism f : A → X one has Image File history File links No higher resolution available. ...

Moreover, any element u∈F(A) defines a natural transformation in this way. The proof in the contravariant case is completely analogous.

In this way, Yoneda Lemma provides a complete classification of all natural transformations from the functor Hom(A,-) to an arbitrary functor F:C→Set.

The Yoneda embedding

An important special case of Yoneda's lemma is when the functor F from C to Set is another hom-functor h_B. In this case, the covariant version of Yoneda's lemma states that

That is, natural transformations between hom-functors are in one-to-one correspondence with morphisms (in the reverse direction) between the associated objects. Given a morphism f : B → A the associated natural transformation is denoted Hom(f,–).

Mapping each object A in C to its associated hom-functor h_A = Hom(A,–) and each morphism f : B → A to the corresponding natural transformation Hom(f,–) determines a contravariant functor h from C to Set^C, the functor category of all (covariant) functors from C to Set. One can interpret h as a contravariant functor: 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. ... For functors in computer science, see the function object article. ...

The meaning of Yoneda's lemma in this setting is that the functor h is fully faithful, and therefore gives an embedding of C^op in the category of functors to Set. The collection of all functors {h_A, A in C} is a subcategory of Set^C. Therefore, Yoneda embedding implies that the category C^op is isomorphic to the category {h_A, A in C}. In category theory, a faithful functor (resp. ...

The contravariant version of Yoneda's lemma states that

Therefore, h′ gives rise to a covariant functor from C to the category of contravariant functors to Set:

Yoneda's lemma then states that any locally small category C can be embedded in the category of contravariant functors from C to Set via h′. This is called the Yoneda embedding.

Preadditive categories, rings and modules

A preadditive category is a category where the morphism sets form abelian groups and the composition of morphisms is bilinear; examples are categories of abelian groups or modules. In a preadditive category, there are both a "multiplication" and an "addition" of morphisms, and that is why preadditive categories are viewed as generalizations of rings. Rings are preadditive categories with one object. A preadditive category is a category that is enriched over the monoidal category of abelian groups. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

The Yoneda lemma remains true for preadditive categories if we choose as our extension the category of additive contravariant functors from the original category into the category of abelian groups; these are functors which are compatible with the addition of morphisms and should be thought of as forming a module category over the original category. The Yoneda lemma then yields the natural procedure to enlarge a preadditive category so that the enlarged version remains preadditive — in fact, the enlarged version is an abelian category, a much more powerful condition. In the case of a ring R, the extended category is the category of all left modules over R, and the statement of the Yoneda lemma reduces to the well-known isomorphism A preadditive category is a category that is enriched over the monoidal category of abelian groups. ... A preadditive category is a category that is enriched over the monoidal category of abelian groups. ... In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ...

M ≅ Hom_R(R,M)   for all left modules M over R.