Yoneda Lemma Generalities Formal Statement Preadditive

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.

This approach is akin to (and in fact generalizes) the common method of studying a ring by investigating the modulesAlgebra In abstract algebra, a module is a generalization of a vector space. In a vector space the set of scalars forms a field whereas in a module the scalars just form a ring. Much of the theory of modules consists of recovering desirable properties of 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.

2 Formal statement

We denote by Fun(C^op,Set) the category of contravariant functorsCategory theory 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. Functor categories are of interest for two main re 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.

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 Mor(Y,A) → Mor(X,A) by the rule f |→ fφ. 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.

This functor is called the Yoneda embedding and it is "natural" in the sense that every functor C → D induces a commutative diagramHomological algebra Category theory 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 obtai

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 bijectionIn 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 . Intuitively, a bijective function creat

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.

3 Preadditive categories, rings and modules

A preadditive categoryA preadditive category is a category that is enriched over the monoidal category of abelian groups. In other words, the category C is preadditive if every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinea is a category where the morphism sets form abelian groupsIn mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed below. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste G and compositions of morphism is bilinear; examples are categories of abelian groups or modules. In a preadditive category, there's 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.

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 right modules over R, and the statement of the Yoneda lemma reduces to the well-known isomorphism

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

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Home > Yoneda lemma

1 Generalities

2 Formal statement

3 Preadditive categories, rings and modules

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