The Yoneda Lemma
By having a product as the domain of a functor, it is possible to have a functor which effectively takes two inputs, so instead of F (( A,B )) we will write F ( A,B ) for ease. The Yoneda Lemma will involve functors C × Set C → Set for any category C . Further, we may denote a component of a natural transformation Φ by Φ A,B rather than Φ ( A,B ) . The Lemma Yoneda’s Lemma is a powerful tool in category theory, and it essentially states that all the structure of a category can be learned by studying its functors to the category of sets—intuitively, if one knows everything about how a category interacts with every other category, one knows everything about it. Before we reach the statement of the theorem, some preliminary definitions will be necessary. Since we seek to show that a category can be embedded in its functor category (to the category of sets), we begin by showing how we can associate to each object a functor. 2 2.1 The hom functor The idea of the hom functor is that we already have a way of transforming two objects into a set, by taking the hom-set. By supplying one object and leaving the other input unfilled (this is known as partial application ), we can create a functor. 4 All that remains is to see where morphisms are taken by the functor. This is effectively asking what the natural way is to associate to each morphism f : X → Y a function Hom( A,X ) → Hom( A,Y ); i.e. how one can use a morphism f : X → Y to associate to each morphism g : A → X a morphism A → Y . The obvious choice for such a morphism is the composition of f with g .
Definition 9 Let C be a category, and A an object in C. The hom functor Hom A : C → Set is given by Hom A ( X ) = Hom( A,X ) for any object X ∈ Ob( C ) , with the morphisms Hom A ( f ) specified by Hom A ( f )( g ) = g ◦ f. 5
Note that we were able to define the morphisms by how they act on elements of their domain. This was possible because we are dealing specifically with the category Set where morphisms are simply functions. It is fairly easy to see that this does indeed define a functor. Clearly Hom A preserves identities, since f ◦ id X will always equal f so the entire hom-set is unchanged by the function Hom A (id X ). Seeing that it preserves composition is similarly simple as Hom A ( g ◦ h )( f ) = f ◦ g ◦ h = (Hom A ( g ) ◦ Hom A ( h )( f ). Thus we have a way of associating to each object A a functor Hom A .
2.2 Statement of theorem The standard statement of the Yoneda Lemma is as follows.
Theorem 1 For any category C, functor F : C → Set , and object A ∈ Ob( C ) , there is an isomorphism between Nat(Hom A ,F ) and F ( A ) , which is natural in A and F.
4 In fact, Hom is already a functor, of a variety known as a bifunctor . Since this is a slightly difficult notion which is not necessary here, I will not discuss it further. 5 The notation Hom A is not standard, but I find it more helpful than the standard notation Hom( A, − ), which can become very confusing.
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