The Yoneda lemma says that for a category , an object in , and a functor , is in bijection with . These bijections are given by and defined as and . This is the covariant form of the Yoneda lemma. There is also a contravariant form that specifies the bijective correspondence . Setting , the Yoneda lemma tells us that natural transformations between the functors and are in bijection with elements of the set .
Let be a monoid, then a left -set is an object in the functor category . Let be a covariant functor then specifies an object in , the image of the unique object of under , together with an endomorphism for each morphism of (i.e. an element of the monoid ) that maps such that and . Similarly, a contravariant functor is a right -set (or equivalently, an -set), but must instead satisfy . When we refer to “-sets” we mean left -sets.
The monoid itself is an -set. Since , viewed as a category, has only a single object, the set is the set of -set endomorphisms of in and so . Note that the set inherits the monoid structure from the properties of composition of morphisms in a category. The covariant functor is a -set (where is the forgetful functor) with acting on by left multiplication since for , is defined by for all . Similarly, the contravariant functor is a -set with acting on by right multiplication. As a result, we denote the -set as .
Consider an -set , then a natural transformation is an -set homomorphism from to . The map is the unique component of the natural transformation. The naturality condition is satisfied since is a -set homomorphism, that is, for a morphism in ,
for all .
Now, we can form the set which consists of morphisms of -sets (i.e. -set homomorphisms). These morphisms are natural transformations of the functors and , so
since we can represent with the functor as every monoid acts on itself by left or right multiplication. Hence by the Yoneda lemma,
We have proved the following result,
-set homomorphisms correspond bijectively to elements of .
This bijection is given by the maps and defined by
They are inverse to each other since
We can interpret this bijection as follows, the choice of determines every other value for since
as is an -set homomorphism.
Cayley’s Theorem arises as a special case of the above result. For a group , set then we have that
and since the set contains automorphisms of , it is a subgroup of the group of all permutations on .
There is a counterpart to the bijection in linear algebra. Consider a field and a -vector space , then . This bijection is given by defined as . The map is clearly linear and injective. To see that it is surjective, notice that for any , there is a map that maps the multiplicative identity onto , namely, . The inverse map is given by , and we have that
However, we are not able to obtain this result from the Yoneda lemma as it has been formulated here. A stronger result, the enriched Yoneda lemma, is required.
Let be a category, then the Yoneda embedding is the functor that maps every object in to the functor and every morphism in to the natural transformation , where each component of the natural transformation is defined by for each object in (i.e. postcomposition with ). The Yoneda embedding is fully faithful which follows from the Yoneda lemma. This means that the functor preserves the relationships that shares with every other object in and so fully faithfully embeds into . That is, for every natural transformation there is exactly one morphism such that . Using this embedding, we are able to work with any category by embedding it in which, in general, has `nicer’ properties and is easier to work with.
We are able to obtain the Yoneda embedding by using the Yoneda lemma in the case where for some object in since the Yoneda lemma says that is in bijection with , which is exactly the Yoneda embedding which assigns to each morphism in the natural transformation in .
Since the Yoneda embedding fully faithfully embeds into , we have that, for any two objects and in ,
and hence, there is a relationship that allows us to characterise an object in by its relationships to all other objects in . That is, the properties of any object in are encoded in the sets for each in .
Let be the monoid of all maps (not just monoid homomorphisms) and map each to the map defined by . Let be the Yoneda embedding of into . Then and since is a single object category. Now, since is fully faithful, the induced functor,
is bijective. But, and
since is the -set which is viewed as a right -set. So we have that the induced functor
defined by , is bijective onto . However, it is only injective into . Now, since , is an injective homomorphism of monoids. The image of is the image of the induced functor which is just the set of all -set homomorphisms from to (or equivalently, right -set homomorphisms from to ).