Yoneda’s Lemma (part 1)

So I think we’re ready, at least for the statement of Yoneda’s lemma. It says that for any locally small category $\mathcal C$ , if is an object in , and $F:\mathcal C\to\textsc{Set}$ a functor, then

$\mbox{Nat}(h^A,F)\cong F(A)$

Moreover, the isomorphism is natural in both and .

Wow that looks complicated. Let’s parse some of the notation that I haven’t even explained yet. So certainly we know what the righthand-side means. It’s applied to . That’s just a set. As for the lefthand-side, $\mbox{Nat}$ and h^A I haven’t explained.

So h^A is a functor we mentioned briefly, but I used a different notation. It’s the representable functor $\hom(A,-)$ . I write it as h^A here to avoid over using parentheses and therefore complicating this business well beyond it’s current level of complication. As a reminder, $\hom(A,-)$ is a functor from $\mathcal C$ to $\textsc{Set}$ (the same as ). It takes objects in $\mathcal C$ to the set of homomorphisms $\hom(A,B)$ . Remember we’re in a locally small category, so $\hom(A,B)$ really is a set. It takes morphisms $f:B\to C$ to a map $h^A(f):\hom(A,B)\to\hom(A,C)$ by sending

$h^A(f):\phi\mapsto f\circ\phi$

We checked all the necessary details in an earlier post to make sure this really was a functor.

So the last thing is that $\mbox{Nat}$ . It’s the collection of all natural transformations between the functors h^A and . So Yoneda’s lemma claims that there is a one to one correspondence between the the natural transformations from h^A to and the set F(A) .

In particular, it claims that $\mbox{Nat}(h^A,F)$ is a set. This is because $\textsc{Set}$ is a locally small category, and so each natural transformation (defined as a collection of morphisms, each of which was a set) is a set. But there are only so many collections of morphisms, not even all of which are natural transformations. The collection is small enough to be a set. If you don’t care about this set theory business. Then disregard the paragraph you probably just read angrily.

It’s worth mentioning now, that Yoneda’s lemma is a generalization of some nice theorems. We can (and will) use it to derive Cayley’s theorem (every group embeds into a symmetric group). We can (and will) use it to derive the important fact that $\hom_R(R,M)\cong M$ in the category of -modules. I bet in that one you can already start to see the resemblance.

We’ll prove Yoneda’s lemma over the next few posts.

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