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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... 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 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 abstract algebra, especially commutative algebra, with geometry. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ...

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 the category we understand best, 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. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ...

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 similar (but not identical) properties to those familiar from the integers. ... In abstract algebra, a module is a generalization of a vector space. ...

Formal statement

We denote by Fun(C^op,Set) the category of contravariant functors from C to Set (Here C^op is the opposite category of C). The morphisms in this category are natural transformations; we will write Nat(F,G) for the set of all natural transformations from the functor F to the functor G. 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 category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in... 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. ...

If A is an object of C, then we can assign to every object X of C the set of morphisms Mor(X,A). Every morphism φ : X → Y in C induces a map . We have thus defined a contravariant functor Mor(-, A) from C to Set, i.e., an element of Fun(C^op,Set). Such a functor is called a representable functor for C; often denoted h_A. In category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. ...

The covariant functor

is called the Yoneda embedding and it is "natural" in the sense that every functor C → D induces a 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. ...

of the corresponding Yoneda embeddings.

The content of the Yoneda lemma is that Y is indeed a full embedding, i.e., for all objects A, B in C, the functor Y induces a bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ...

In other words: the morphisms between A and B in the original category C are "the same" as the ones between the two corresponding objects Y(A), Y(B) in the extended category Fun(C^op,Set).

And even more: for any contravariant functor F : C → Set and for any object A in C, there is a natural bijection

which means that, if you know how the functor F behaves on C, then you also know how it relates to the image of C in the extended category.

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's 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, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...

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 certain kind of category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ... In abstract algebra, a module is a generalization of a vector space. ...

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