How bewildering is the Yoneda embedding?

There’s now another one-chapter update for Category Theory II.

There’s some minor earlier tinkering, but Chapter 37 has been considerably revised. The proofs leading up to what I call the Restricted Yoneda lemma and the Yoneda Embedding Theorem have been tidied up, and should be much clearer. And the final section “Yoneda meets Cayley” — which was a mess, almost incoherently so — is now crisp and clear. I hope!

Tom Leinster has written “The level of abstraction in the Yoneda Lemma means that many people find it quite bewildering.” While Awodey calls it “the single most used result” of category theory. So: bewildering but centrally important?

Well, I really do hope the decaffeinated version of Yoneda in Chapter 37 really is plain sailing. There’s basically one small idea — you can use a \mathsc{C}-arrow f\colon B \to A very simply to construct a natural transformation between hom-functors \mathsc{C}(A, --) and \mathsc{C}(B, --) — and then all the rest is pretty much applying definitions in obvious ways. So far, I hope, not bewildering at all!

1 thought on “How bewildering is the Yoneda embedding?”

  1. When I first found bewildering back in my undergrad days was that the intuition behind the Yoneda lemma was usually given in textbooks as something like “objects only depend on the mappings to/from them”, which is very nice, but it only applies to what you call Theorem 177, which is indeed what most often gets used about Yoneda. I found that the “full” Yoneda lemma (of your Chapter 38) is not fully captured by this intuition, but the textbooks pretended that it is, which caused some confusion back then. (I still haven’t found a good intuition for that one.)

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