Bewildered of Cambridge writes …

A few posts back, I quoted Tom Leinster: “The level of abstraction in the Yoneda Lemma means that many people find it quite bewildering.” I can relate to that. Just a couple of days ago I managed to bewilder myself for a long afternoon, and it took me an embarrassing amount of time to locate where I’d composed arrows the wrong way around (contravariance bites back). I have a rule when writing this sort of expository stuff not to look things up, on the principle that working things out again for myself from first principles much increases the chances of being able to explain the topic lucidly to others. But the rule has its downside sometimes!

Anyway …

There is now a fifteen page chapter on Yoneda (the theorem, not its implications which I plan to tackle next). I take things in three stages, proving what I (non-standardly) call the Restricted Yoneda Lemma, and then the Intermediate Yoneda Lemma, before getting to the fully caffeinated Yoneda Lemma. The hope is that, chunked up like this,  each of the three stages almost writes itself …

For enthusiasts, the latest version of Category Theory II is here.

3 thoughts on “Bewildered of Cambridge writes …”

  1. I think Leinster’s remark needs unpacking. Why would a (presumably high) level of abstraction make something bewildering? And how is it more abstract than lots of other things? When I look at the Wikipedia page on the Yoneda lemma, ok, sure, it eventually becomes bewildering, but that’s because it gets into details involving other category-theoretic concepts and expresses things in formulas that someone not already familiar with the things involved will find quite cryptic (to say the least).

    The nLab page is more immediately bewildering because it begins like this:

    The Yoneda lemma says that the set of morphisms from a representable presheaf y(c) into an arbitrary presheaf X is in natural bijection with the set X(c) assigned by X to the representing object c.

    So it immediately suggests “I need to know a bunch of other things before I can understand this”.

    My theory is, therefore, that the problem is not that the lemma is very abstract but that it’s at the top of a tower of other abstractions; so not just any old way of being highly abstract: that particular way.

    1. I wasn’t taking Leinster’s remark too seriously …

      I think, however, that some of the textbook proofs I’ve seen do engender a possibly befuddling sense of mystery … hence my effort to do a bit better.

      1. I don’t think Leinster was talking about the proofs being bewildering, though. It’s possible he’s said it more than once, I suppose. Still, one place where he said it is in his The Yoneda Lemma: What’s It All About?. That paper doesn”t explain the proof. It even says:

        The proof was in the lectures so I won’t reproduce it here; in any case once you have thoroughly understood the statement, you should find the proof straightforward.

        Here you’re saying “some of the textbook proofs I’ve seen do engender a possibly befuddling sense of mystery.” Your Yoneda Without (Too Many) Tears is also primarily about the proofs. And now, when I look back at the earlier Logic Matters post How bewildering is the Yoneda embedding?, it also seems to be largely about the proofs. That isn’t what I thought when I first read it, and it’s not what Andrei’s comment there is about either.

        His comment is about the sort of thing I most want explained. I might be interested in the proofs once I have a better understanding of what the Lemma is about and why it’s interesting or useful (especially if the use or interest extends outside category theory). Until then, the proofs are just a lot of technical detail with no clear payoff to slogging through them.

        Here’s his comment for easy reference:

        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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