Yoneda lemma
In mathematics, the Yoneda lemma is a fundamental result in category theory.
[1] 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 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.
The Yoneda lemma suggests that instead of studying the locally small category
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.
Yoneda's lemma concerns functors from a fixed category
is a locally small category (i.e. the hom-sets are actual sets and not proper classes), then each object
This version involves the contravariant hom-functor which sends
to the map This is enough to determine the other functor since we know what the natural isomorphism is.
Many texts and articles either use the opposite convention or completely unrelated symbols for these two functors.
However, most modern algebraic geometry texts starting with Alexander Grothendieck's foundational EGA use the convention in this article.
The proof in the contravariant case is completely analogous.
[1] An important special case of Yoneda's lemma is when the functor
In this case, the covariant version of Yoneda's lemma states that That is, natural transformations between hom-functors are in one-to-one correspondence with morphisms (in the reverse direction) between the associated objects.
The contravariant version of Yoneda's lemma states that Therefore,
: Yoneda's lemma then states that any locally small category
The Yoneda embedding is sometimes denoted by よ, the hiragana Yo.
[4] The Yoneda embedding essentially states that for every (locally small) category, objects in that category can be represented by presheaves, in a full and faithful manner.
That is, for a presheaf P. Many common categories are, in fact, categories of pre-sheaves, and on closer inspection, prove to be categories of sheaves, and as such examples are commonly topological in nature, they can be seen to be topoi in general.
The Yoneda lemma provides a point of leverage by which the topological structure of a category can be studied and understood.
[6] A preadditive category is a category where the morphism sets form abelian groups and the composition of morphisms is bilinear; examples are categories of abelian groups or modules.
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.
, and the statement of the Yoneda lemma reduces to the well-known isomorphism As stated above, the Yoneda lemma may be considered as a vast generalization of Cayley's theorem from group theory.
A natural transformation between such functors is the same thing as an equivariant map between
on itself by left-multiplication (the contravariant version corresponds to right-multiplication).
But it is easy to see that (1) these maps form a group under composition, which is a subgroup of
Yoshiki Kinoshita stated in 1996 that the term "Yoneda lemma" was coined by Saunders Mac Lane following an interview he had with Yoneda in the Gare du Nord station.
