The Yoneda Lemma
Cayley’s theorem states that every group can be embedded in a symmetric group. Though I have not the space to show this formally, I provide a sketch of a proof. The main idea is that, as the second functor F , we take again Hom • , giving us an isomorphism between Nat(Hom • , Hom • ) and Hom(• , •). The right hand side is the entire group (since it has no structure other than the hom-sets on its single object), while the left hand side is a set of natural transformations. However, each natural transformation Φ has only a single component Φ Hom( • , •) with which it can be identified, yielding a set of functions Φ : Hom(• , •) → Hom(• , •). Since the naturality condition along with the functorial nature of Hom essentially tells us the structures have been preserved, •we know that Nat(Hom , Hom ) is also a set containing the morphisms of a group, so each Φ is an isomorphism. Thus, the left-hand side is a subset of S Hom( • , •), proving (intuitively) the theorem. For a more detailed overview of group theory, along with a formal proof of Cayley’s Theorem, I have found [1] helpful; it is old enough that an interested reader can make their own attempt at seeing how various results have categorical representations, without being told. 3.2 Enriched categories In the previous section I gave a taste of how category theory can be applied. Now, I sketch how the Yoneda Lemma generalizes to the even more abstract setting of an enriched category . The fundamental idea is that one need not necessarily have hom-sets: for example, a hom-group (where there is an additional way of composing morphisms) is a sensible idea. We observe that the composition in an ordinary category is essentially a function ◦ : Hom( A,B ) × Hom( B,C ) → Hom( A,C ), and that the notion of function is a special case of morphisms, namely in the case of Set . Similarly, the identity id A ∈ Hom( A,A ) can be specified as a function id : 1 → Hom( A,A ) from a single-element set 1 , since a function from a one-element set picks out a single element of its codomain. Thus, in order to generalize categories with hom- sets which are elements of Set to enriched categories with hom- objects which are elements of another category V , we need only that this category V should have the concepts of singleton and product, which we generalize to monoidal unit and monoidal product . This leads to the notion of a monoidal category . Ordinary categories can thus be seen as categories enriched in Set . In the context of enriched categories, the hom functor maps to V , not Set , so the Yoneda Lemma concerns other functors F : C → V . 7 Its statement, though much more complicated than the case I have shown, follows the same intuition, allowing any category to be embedded in a functor category; an exploration of enriched categories together with a general proof can be found at [5]. 4 Conclusion This concludes my exploration of the Yoneda Lemma. I have introduced many of the key concepts of category theory, and alluded to several others along the way; I hope the reader has learned something from this. I regret that this foray into the categorical must be so brief: there are fundamental concepts like duality, limits, and categorification that I have not had the space to mention. An interested reader must now do their own research, but they may do so knowing that they have seen the important result that is the Yoneda Lemma proven formally.
7 A meticulous reader may note that V is an ordinary category enriched in Set , not in V , and so it is unclear how such a functor would be defined. In fact things are somewhat more complicated, and such a hom functor is only allowed in categories that have an internal hom , meaning that hom-objects are themselves part of the same category: for example, hom-sets in Set are themselves in Set .
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