Yoneda lemma

In mathematics, the Yoneda lemma is arguably the most important result in . It 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 miniature category with just one object and only isomorphisms). It allows the of any into a (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. The Yoneda lemma suggests that instead of studying the locally small category , one should study the category of all functors of into (the with functions as morphisms). is a category we think we understand well, and a functor of into can be seen as a "representation" of in terms of known structures. The original category is contained in this functor category, but new objects appear in the functor category, which were absent and "hidden" in . 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 , and the category of modules over the ring is a category of functors defined on . Yoneda's lemma concerns functors from a fixed category to the , . If is a (i.e. the hom-sets are actual sets and not proper classes), then each object of gives rise to a natural functor to called a hom-functor. This functor is denoted: The (covariant) hom-functor sends to the set of morphisms and sends a morphism (where and are objects in ) to the morphism (composition with on the left) that sends a morphism in to the morphism in . That is, Yoneda's lemma says that: Let be a functor from a locally small category to .