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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 and only isomorphisms). 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.

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GeneralitiesEdit

The Yoneda lemma suggests that instead of studying the (locally small) category  , one should study the category of all functors of   into   (the category of sets 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  .

Formal statementEdit

Yoneda's lemma concerns functors from a fixed category   to the category of sets,  . If   is a locally small category (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   to the morphism   (composition with   on the left) that sends a morphism   in   to the morphism   in  . That is,

 
 .

Let   be an arbitrary functor from   to  . Then Yoneda's lemma says that:

For each object   of  , the natural transformations from   to   are in one-to-one correspondence with the elements of  . That is,

 .

Moreover this isomorphism is natural in   and   when both sides are regarded as functors from   to  . (Here the notation   denotes the category of functors from   to  .)

Given a natural transformation   from   to  , the corresponding element of   is  ;[a] and given an element   of  , the corresponding natural transformation is given by  .

There is a contravariant version of Yoneda's lemma, which concerns contravariant functors from   to  . This version involves the contravariant hom-functor

 

which sends   to the hom-set  . Given an arbitrary contravariant functor   from   to  , Yoneda's lemma asserts that

 

Naming conventionsEdit

The use of   for the covariant hom-functor and   for the contravariant hom-functor is not completely standard. 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.[b]

The mnemonic "falling into something" can be helpful in remembering that   is the contravariant hom-functor. When the letter   is falling (i.e. a subscript),   assigns to an object   the morphisms from   into  .

ProofEdit

The proof of Yoneda's lemma is indicated by the following commutative diagram:

This diagram shows that the natural transformation   is completely determined by   since for each morphism   one has

 .

Moreover, any element   defines a natural transformation in this way. The proof in the contravariant case is completely analogous.

The Yoneda embeddingEdit

An important special case of Yoneda's lemma is when the functor   from   to   is another hom-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. Given a morphism   the associated natural transformation is denoted  .

Mapping each object   in   to its associated hom-functor   and each morphism   to the corresponding natural transformation   determines a contravariant functor   from   to  , the functor category of all (covariant) functors from   to  . One can interpret   as a covariant functor:

 

The meaning of Yoneda's lemma in this setting is that the functor   is fully faithful, and therefore gives an embedding of   in the category of functors to  . The collection of all functors   is a subcategory of  . Therefore, Yoneda embedding implies that the category   is isomorphic to the category  .

The contravariant version of Yoneda's lemma states that

 

Therefore,   gives rise to a covariant functor from   to the category of contravariant functors to  :

 

Yoneda's lemma then states that any locally small category   can be embedded in the category of contravariant functors from   to   via  . This is called the Yoneda embedding.

Representable functorEdit

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, pre-sheaves, and on closer inspection, prove to be 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.

Preadditive categories, rings and modulesEdit

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. In a preadditive category, there is both a "multiplication" and an "addition" of morphisms, which is 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  , the extended category is the category of all right modules over  , and the statement of the Yoneda lemma reduces to the well-known isomorphism

    for all right modules   over  .

Relationship to Cayley's theoremEdit

As stated above, the Yoneda lemma may be considered as a vast generalization of Cayley's theorem from group theory. To see this, let   be a category with a single object   such that every morphism is an isomorphism (i.e. a groupoid with one object). Then   forms a group under the operation of composition, and any group can be realized as a category in this way.

In this context, a covariant functor   consists of a set   and a group homomorphism  , where   is the group of permutations of  ; in other words,   is a G-set. A natural transformation between such functors is the same thing as an equivariant map between  -sets: a set function   with the property that   for all   in   and   in  . (On the left side of this equation, the   denotes the action of   on  , and on the right side the action on  .)

Now the covariant hom-functor   corresponds to the action of   on itself by left-multiplication (the contravariant version corresponds to right-multiplication). The Yoneda lemma with   states that

 ,

that is, the equivariant maps from this  -set to itself are in bijection with  . But it is easy to see that (1) these maps form a group under composition, which is a subgroup of  , and (2) the function which gives the bijection is a group homomorphism. (Going in the reverse direction, it associates to every   in   the equivariant map of right-multiplication by  .) Thus   is isomorphic to a subgroup of  , which is the statement of Cayley's theorem.

HistoryEdit

Yoshiki Kinoshita stated in 1996 that the term "Yoneda lemma" was coined by Saunders Mac Lane following an interview he had with Yoneda.[1]

See alsoEdit

NotesEdit

  1. ^ Recall that   so the last expression is well-defined and sends a morphism from   to  , to an element in  .
  2. ^ A notable exception to modern algebraic geometry texts following the conventions of this article is Commutative algebra with a view toward algebraic geometry / David Eisenbud (1995), which uses   to mean the covariant hom-functor. However, the later book The geometry of schemes / David Eisenbud, Joe Harris (1998) reverses this and uses   to mean the contravariant hom-functor.

ReferencesEdit

  1. ^ Kinoshita, Yoshiki (23 April 1996). "Prof. Nobuo Yoneda passed away". Retrieved 21 December 2013.

External linksEdit