Theorems implied by Yoneda's lemma?

Ok, so I was reading the Wikipedia article on Yonedas lemma. And I’ve heard before that when you prove things in category theory you automatically get a lot of results by proving it in abstract categories.

One of my favorite things in math is the whole Lagrange’s theorem $\rightarrow$ Euler’s theorem $\rightarrow$ Fermat’s little theorem chain.

Are there examples of other “semi-mainstream” theorems trivialized by Yoneda?


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The most beautiful and yet elementary things I noticed when I was learning the basics were about Yoneda Lemma; these are not “theorems” in the real sense of the term; instead they are principles one sees in action when proving things.

  1. There this thing called Yoneda lemma where you prove that given a contravariant functor $\mathcal C\to \bf Set$, the correspondence $F\mapsto \text{Nat}(\hom(-,A),F)$ “acts as an evaluation” giving you a set with as much elements as $FA$ back. Ok then, let’s play this game where we evaluate at the object $B$ the functor $\hom(-,B)$: we get that

$\text{Nat}(\hom(-,A),\hom(-,B))\cong \hom(A,B)$, i.e. that the correspondence which sends an object to its representable presheaf is fully faithful.

  1. Yoneda Lemma allows you to reduce statements about complicated categories to statements about sets, or better to say, functors which take value in $\bf Set$. This is because

the previous point gives you a fully faithful functor $\mathcal C \to [\mathcal C^\text{op},\bf Set]$, the Yoneda embedding $A\mapsto \hom(-,A)$;

  1. there’s this thing you maybe learned about a “product of $A,B$” in a category being an object $P$ endowed with two maps $A\leftarrow P\to B$ such that blablabla; ok then, let’s feed the Yoneda representable functor $\hom(X,-)$ with this diagram: we obtain something which tells you that $\hom(X,P)$ is precisely the product (i.e. the set theoretic product, the one you learn in your first day as a freshman in any “calculus 0” course) of $\hom(X,A)$ and $\hom(X,B)$. This gives you that

$P\cong A\times B$ in a category $\cal C$ if and only if for any $X\in\cal C$ the set $\hom(X,P)$ is in bijection the product of $\hom(X,A)$ and $\hom(X,B)$, and this bijection is natural, ie. respects the fact that I can have arrows $Y\to X$, generating arrows $\hom(X,P)\to \hom(Y,P)$.

The same argument (even if it’s far more involved to grasp “visually”) works well for any shape of diagram: equalizers/kernels, pullbacks, inverse limits … and properly dualized, it works well with colimits!

Whenever you are able to characterize a universal object (i.e. a limit/colimit) in the category of sets, then you can define that very universal object in any category exploiting the former principle: an object $K$ with maps $A_i\to K$ in $\mathcal C$ is universal (say, a colimit for the diagram $F\colon i\mapsto A_i$) if and only if passing its arrows $A_i\to K$ through the yoneda embedding $\hom(-,X)$, I obtain the set theoretic universal (in fact the limit, since $\hom(-,X)$ is contravariant), naturally in $X$.

Everything I said deeply relies on the fact that you are using sets. Or maybe not? There this thing called “enriched Yoneda lemma”, which is

the same statement, but for functors between any $\bf Ab$-enriched category [where each $\hom(X,Y)$ is a $\mathbb Z$-module] and the category of abelian groups…

The Yoneda lemma allows you to prove one of the most useful results in basic category theory, namely that left/right adjoint functors preserve colimits/limits of any shape. I can’t even estimate the number of times this simple remark saved my life in practice. All you have to do is follow a suitable string of isomorphisms, and then say:

Since the yoneda embedding is fully faithful, it reflects isomorphisms, i.e. whenever $\hom(A,X)\cong \hom(B,X)$, or $\hom(X,A)\cong \hom(X,B)$, for any $X\in\cal C$, naturally in $X$, then $A\cong B$.

Cayley’s theorem. Yoneda embedding. Uniqueness of representing objects

The correspondence between characteristic classes on the one hand and cohomology classes on a classifying space on the other is a nice application of Yoneda’s lemma, though it probably doesn’t deserve the name of “theorem”.