Yoneda lemma

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 (contravariant set-valued 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.

Generalities

The Yoneda lemma suggests that instead of studying the (locally 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 think 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.