The newly revised chapter from Category Theory II is now a draft stand-alone PDF, Yoneda Without (Too Many) Tears.
All comments and corrections welcome before I perhaps put a revised version on the arXiv. It took me some effort back in the day to get this clear enough in my own mind; and I have tidied up the presentation, perhaps some others might appreciate a helping hand through the proof!
I feel about this like I do about many things in category theory. I would like an intuitive explanation first, and indication of why it’s interesting or useful, and, in the proofs, explanations of where we’re going and why we’re doing what we’re doing as a way to get there — instead of technical detail after technical detail. The overview at the start of what we’ll be doing is good up to a point (it doesn’t present an intuitive picture of what the whole thing is about), but once we launch into the proofs, it’s technical detail after technical detail, amounting to a proof of Theorem 5 (The Restricted Yoneda Lemma) before we’ve even seen that theorem.
From that POV, things don’t get much better until we reach §6 where, suddenly, they do, and we get a higher level explanation of what we were doing before and why, plus a useful look forward. There’s even a day-brightening “It is forgivable to skip the details”.
But skipping the details, what would we get? What is the reader being offered? Slog through more technical stuff (or skip the details), and …?
At least §6 recognises that something is missing:
And, to be fair, although the title of this article says “Yoneda Without (Too Many) Tears”, it’s soon made clear that the article just “aims to make the proof of the full Yoneda Lemma as unpuzzling as possible — just the proof. So “why do the theorems matter?” is beyond its scope.
In the Notes, though, I’d expect that to come up fairly soon, in a later chapter. It doesn’t, so far as I could tell. Even the word “Yoneda” appears only twice (according to the search facility in Firefox).