Table of contents |

2 Formal statement 3 Preadditive categories, rings and modules |

Generally speaking, the Yoneda lemma suggests that instead of studying the (small) category *C*, one should study the category of all functors of *C* into **Set** (where **Set** is the category of all 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 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*.

Such a functor is called a representable functor for *C* - often denoted h_{A}.

- Y :
*C*→ Fun(*C*^{op},**Set**)

of the corresponding Yoneda embeddings.D-->Fun(D^{op},Set) | | | | | | V VC-->Fun(C^{op},Set)

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*

- Mor(
*A*,*B*) → Nat(Y(*A*), Y(*B*)).

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

*F*(*A*) → Nat(Y(*A*),*F*)

A *preadditive category* is a category where the morphism sets form abelian groups 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*.