This article may be too technical for most readers to understand.(August 2021)
In mathematics, the Yoneda lemma is arguably the most important result in category theory. 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.
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 .
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,
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
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 .
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 embedding
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.
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
For any functors and the following formulas are all formulations of the Yoneda lemma.
Preadditive categories, rings and modules
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
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.
- Recall that so the last expression is well-defined and sends a morphism from to , to an element in .
- 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.
- Freyd, Peter (1964), Abelian categories, Harper's Series in Modern Mathematics (2003 reprint ed.), Harper and Row, Zbl 0121.02103.
- Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics, 5 (2nd ed.), New York, NY: Springer-Verlag, ISBN 0-387-98403-8, Zbl 0906.18001
- Loregian, Fosco (2015). "This is the (co)end, my only (co)friend". arXiv:1501.02503 [math.CT].