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 allows the embedding of any locally small 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 edit

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 statement edit

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   (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:

Lemma (Yoneda) — Let   be a functor from a locally small category   to  . Then 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   which assigns to a morphism   a value of  .

Contravariant version edit

There is a contravariant version of Yoneda's lemma,[2] 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

 

Naturality edit

The bijections provided in the (covariant) Yoneda lemma (for each   and  ) are the components of a natural isomorphism between two certain functors from   to  .[3]: 61  One of the two functors is the evaluation functor

 
 

that sends a pair   of a morphism   in   and a natural transformation   to the map

 

This is enough to determine the other functor since we know what the natural isomorphism is. Under the second functor

 
 

the image of a pair   is the map

 

that sends a natural transformation   to the natural transformation  , whose components are

 

Naming conventions edit

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 covariant hom-functor. When the letter   is falling (i.e. a subscript),   assigns to an object   the morphisms from   into  .

Proof edit

Since   is a natural transformation, we have the following commutative diagram:

 
Proof of Yoneda's lemma

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.[1]

The Yoneda embedding edit

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.

The Yoneda embedding is sometimes denoted by よ, the Hiragana kana Yo.[4]

Representable functor edit

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.

In terms of (co)end calculus edit

Given two categories   and   with two functors  , natural transformations between them can be written as the following end.[5]

 

For any functors   and   the following formulas are all formulations of the Yoneda lemma.[6]

 
 

Preadditive categories, rings and modules edit

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 theorem edit

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.

History edit

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.[7][8]

See also edit

Notes edit

  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.

References edit

  1. ^ a b Riehl, Emily (2017). Category Theory in Context (PDF). Dover. ISBN 978-0-486-82080-4.
  2. ^ Beurier & Pastor (2019), Lemma 2.10 (Contravariant Yoneda lemma).
  3. ^ Mac Lane, Saunders (1998). Categories for the working mathematician. Graduate Texts in Mathematics. Vol. 5 (2 ed.). New York, NY: Springer. doi:10.1007/978-1-4757-4721-8. ISBN 978-0-387-98403-2. ISSN 0072-5285. MR 1712872. Zbl 0906.18001.
  4. ^ "Yoneda embedding". nLab. Retrieved 6 July 2019.
  5. ^ Loregian (2021), Theorem 1.4.1.
  6. ^ Loregian (2021), Proposition 2.2.1 (Ninja Yoneda Lemma).
  7. ^ Kinoshita, Yoshiki (23 April 1996). "Prof. Nobuo Yoneda passed away". Retrieved 21 December 2013.
  8. ^ "le lemme de la Gare du Nord". neverendingbooks. 18 November 2016. Retrieved 2022-09-10.

External links edit