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The Yoneda lemma in Category Theory
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When I first learned of Category Theory I was awestruck with how beautifully it captured the commonality of so many structures of mathematics. I thought at the time that the theorems of Category Theory would be such that one could see how they were manifested in various categories. That is to say, Theorem A of category theory might be seen to take the form of Theorem B in the category of Groups and Theorem C in the category of Topologies. But the development of Category Theory did not take this form. Instead the theorems of Category Theory take the form of mind-numbingly abstract propositions in pure Category Theory. The Yoneda Lemma is one of those propositions.

Before dealing with the Yoneda Lemma itself it is fruitful to consider Cayley's Theorem for Groups. It is said that Yoneda Lemma is vast generalization of Cayley's Theorem.

Cayley's Theorem for Groups

Cayley's Theorem: Any group is isomorphic to a subgroup of a permutations group.

Arthur Cayley was an Irish mathematician. The name Cayley is the Irish name more common spelled Kelly.

Proof:

Let S be the set of elements of a group G and let * be its operation. Let F be the set of one-to-one functions from the set S to the set S. Such functions are called permutations of the set. The set F with function composition (·) is a group.

Proof of this proposition:

Function composition is closed and associative. There is an identity element e(x)=x for all x belonging to S. There is an inverse for any function: if f(x)=y then f-1(y)=x. Thus (F, ·) is a group.

For any element g of S consider the function fg(x)=g*x for all x in S. This function is an element of F.

Consider fg*h(x). Since G is a group g*h is an element of S and hence fg*h is an element of F. Furthermore, since * is associative in G,

(g*h)*x = g*(h*x) = g*(fh(x)) = fg(fh(x)) = fg·fh(x)
but
(g*h)*x is fg*h(x)
so
fg*h = fg·fh

Therefore the set {fg for g in G} is a subgroup of F. G is isomorphic to a subgroup of F with (·).

Definitions Essential to the Yoneda Lemma

Homomorphism: A homomorphism is a map between two algebraic structures that preserves selected properties.

Isomorphism: An isomorphism is a homomorphism that has an inverse.

The Yoneda Lemma

The Yoneda lemma indicates that instead of studying a locally small category C, one can fruitfully study the category of all functors of C into Set (the category of sets with functions as morphisms). Set is a familiar category and a functor of C into Set can be seen as a "representation" of C in terms of the known structures of Set. The original category C is contained in this functor category, but new objects appear in the functor category, which were absent and "hidden" in C. Treating these new objects just like the old ones often unifies and simplifies the theory.

Yoneda's lemma concerns functors from a fixed category C to the category of sets, Set. If C is a locally small category, meaning its sets of Objects are actual sets rather than proper classes, then each object A of C gives rise to a natural functor to Set called a hom-functor. This functor is denoted:

hA(X) = Hom(A,X).

The (covariant) hom-functor hA(X) sends X to the set of morphisms Hom(A,X) and sends a morphism f:X →Y to the morphism f o - (composition with f on the left) that sends a morphism g in Hom(A,X) to the morphism f o g in Hom(A,Y). That is,

f → {Hom}(A,f) = [Hom}(A,X) ∋ g → f o g ∈ {Hom}(A,Y) \] .

Let F be an arbitrary functor from C to Set. Then Yoneda's lemma says that:

For each object A of C, the natural transformations from hA to F are in one-to-one correspondence with the elements of F(A). That is,

Nat}(hA,F) ≅ F(A).

Moreover this isomorphism is natural in A and F when both sides are regarded as functors from SetC x C to Set. (Here the notation SetC denotes the category of functors from C to Set.)

Given a natural transformation Φ from hA to F, the corresponding element of F(A) is u =

(To be continued.)

Sources:

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


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