Yoneda lemma

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In mathematics, specifically in category theory, the Yoneda lemma 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 (viewing a group as a particular kind of 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.

Contents

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.

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.

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 gives rise to a natural functor to Set called a hom-functor. This functor is denoted:

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