Once upon a lifetime ago, I took Part III of the Maths Tripos.
In fact, rather alarmingly, I started exactly fifty years ago this term. And it was tough. You had to aim to do over the year (the equivalent of) six courses of 24 lectures, which were lectured at a helter-skelter, take-no-prisoners, pace. The blackboard notes gave you just the barest skeleton, and you had to spend a great deal of time working on them between classes in order to keep up, and then a lot more time in the vacations to really get on top of the material. I remember it as the time in my life I had to work by far the hardest, though it all worked out well.
Things, it seems, have changed astonishingly little. I’ve been turning out — Mondays, Wednesdays and Fridays at 9! — to go to this year’s Part III Category Theory lectures (given by Rory Lucyshyn-Wright. The course is still lectured at a cracking pace, with blackboard notes giving you a bare skeleton, and leaving a great deal of work required if you are to put enough flesh onto the bones to get the real shape of what’s going on. No pre-digested handouts here!
I’m just about hanging on in there. I’m trying to write up quite detailed notes to fix ideas, and I’m already falling behind with those — and this despite the fact that I’ve read around a bit the subject in the past. But, as we all know, in maths in particular there is all the difference between a casual read and really working your way into a topic. And that’s what I want to try to do, at least for the beginnings of category theory. (Well, why not?)
OK, I’m no doubt slower on the uptake than I was back in the day, and the kids around me are among the world’s best mathematicians of their age, have a lot more energy and function more hours in the day. But they are having to keep up with three times as much this term, and will do it all again next term. We can only be impressed.
Speaking about category theory: as I see it, there is a kind of (very successful) paradigm in contemporary mathematics that tells us that every kind of mathematics is in the end reducible to logic plus set theory (ZFC etc.). So all of mathematics should, in principle, be derivable from formal logic plus the axioms of set theory. I always wondered if this is true also for category theory, being a kind of meta-theory that speaks, among other things, about the category of sets, of groups etc. Is category theory just another consequence of set theory? Or can category theory itself serve as the foundation of mathematics, instead of set theory?
Yours are exactly the questions I’d like to get clearer about myself! Hence my late developing interest in category theory. That are some relevant remarks and a lot of references in the Stanford Encyclopedia article http://plato.stanford.edu/entries/category-theory/
Leinster’s “An informal introduction to topos theory” (http://arxiv.org/abs/1012.5647) can answer your question well.
It requires some category theory to appreciate in full, but even without knowing any category theory, I’ll claim that you can read the introduction and the relevant parts of Sec. 2 (which describe why Lawvere’s Elementary Theory of the Category of Sets is cool). And the categorical requirements for Sec. 1 and 2 are limited.