Work on the Gentle Intro to Category Theory has been going slowly for a couple of months. I’d got to the point where I needed to backtrack and think through what I wanted to say, in a couple of preliminary chapters, about sets and then about structures, before going on to talk about category theory as (inter alia) a framework for talking about how structures-of-structures hang together.
The chapter on sets, in particular, has been causing me grief. This isn’t going to be one of those merely routine chapters about basic set theoretic notions and notations — I’m going to take it that anyone tackling even an intro on category theory is unlikely to need yet another run through those basics. Nor is this going to be a discussion of the fancy issues about big universes of sets that might or might not be needed for really big categories (after all, it’s only after we know quite a bit of category theory that it will become clear what the genuine issues are and how much of applicable category theory they affect).
No, what I want to write about are basic things that turn out to be relevant quite soon when we start talking about categories, like how much common-or-garden maths-talk seemingly about sets is really plural talk in thin disguise (and why plural talk should be taken at face value), and relatedly about how much set talk is about virtual classes in Quine’s sense. And then say something about what we get from having a more substantial set theory, and just how much substantial set theory is needed for ‘ordinary’ (non-set-theoretic) maths, and why there is no unique theory which delivers the goods. Fundamental but not-so-simple things like that, familiar to logicians but not necessary to mathematicians (who need to be convinced why, here at least, the should care)! I think I’m beginning to see how to shape such a chapter to say what needs to be said without getting too tendentious, after some fun reading around for more inspiration. And since I couldn’t see that a few weeks ago, there must have been progress, even if at a snail’s pace.
I’ll put the draft chapter online for comments when it’s done. But some very nice forthcoming events here mean I have to put further work on hold for a bit. So don’t hold your breath.