The Yoneda Lemma
A significant advantage of category theory is that the theorems and definitions are often very concise. However, this sometimes comes at the cost of a lack of clarity for one who is unfamiliar with the language. Here, I find the statement that an isomorphism is ‘natural in A and F ’ is rarely explained adequately. What it means is that the functor LHS : C × Set C → Set defined by LHS ( A,F ) = Nat(Hom A ,F ) is naturally isomorphic to the functor RHS given by RHS ( A,F ) = F ( A ). For obvious reasons, the functor I call RHS is often called the application functor . 6 A further confusion from this definition is that how the functors LHS and RHS act on morphisms is not specified. We shall see that there is a natural choice, but the theorem only states that some functors that produce the results given as objects are isomorphic; we shall prove this by construction. 2.3 The bijection We asserted that Nat(Hom A ,F ) is naturally isomorphic to F ( A ). To show this, we must first show that they are indeed isomorphic. Since we are working in Set , an isomorphism is simply a bijection; i.e. a 1–1 mapping between the sets. Thus we seek a way to associate to each natural transformation Φ : Hom A → F a unique element x ∈ F ( A ). Intuitively, the ideal way of constructing our bijection would be to pick out a component of Φ which has some special significance, then to apply it to some element of its domain which is somehow special. In fact we can proceed by doing exactly that. We first observe that at least one component of Φ has F ( A ) as its codomain, namely Φ A . In general we cannot guarantee that this is the only such component since we could have some object B with F ( A ) = F ( B ), but Φ A is the only choice we can guarantee will have the correct codomain in all cases. The domain of Φ A is then Hom A ( A ), i.e. Hom( A,A ), the set of morphisms from A to itself. Of these, the obvious choice for a ‘special’ morphism is id A . Thus, we suggest that the isomorphism we seek, which we shall call α , associates to each natural transformation Φ the object α (Φ) = Φ A (id A ) ∈ F ( A ). 2.3.1 Proof It now remains to show that the above function α : Nat(Hom A ,F ) → F ( A ) is indeed bijective. Let X ∈ Ob( C ) be any object and f : A → X any function between the specified hom-sets; let Φ : Hom A → F be a natural transformation. The naturality condition on Φ (from Definition 5) tells us that Hom A ( f ) ◦ Φ X = Φ A ◦ F ( f ). Since both sides of this equality are morphisms in Set (i.e. functions), we can apply them to id A to yield
(Hom A ( f ) ◦ Φ X )(id A ) = (Φ A ◦ F ( f ))(id A )
Applying the definition of function composition from Definition 2 and how hom functors treat morphisms (see Definition 9), we see that
Φ X (Hom A ( f )(id A )) = F ( f )(Φ A (id A )) = Φ X (id A ◦ f ) = Φ X ( f ) = F ( f )( α (Φ))
We now observe that a natural transformation Φ is specified entirely by the collection of Φ X ( f ) for all X and f , since Φ X is a function (any function can be specified entirely by how it acts on all inputs in its domain), and a natural transformation is specified entirely by its components. Since F and f are both independent of Φ, we can intuitively see from this identity that Φ is specified entirely by α (Φ), suggesting α is indeed our isomorphism. Formalizing this is not too hard: we just state α − 1 explicitly.
6 The names LHS and RHS are of no import: they stand for left-hand side and right-hand side.
75
Made with FlippingBook - PDF hosting