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 A is an object in C, 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 A and F.

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 F applied to A. 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 F). It takes objects B 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 F. So Yoneda’s lemma claims that there is a one to one correspondence between the the natural transformations from h^A to F 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 R-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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One Response to Yoneda’s Lemma (part 1)

  1. Pingback: Applying Yoneda’s Lemma « Andy Soffer

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