Thu, 7th Oct 2010 (CountingOnMyFingers)
Teaching Category Theory
There was a question on MO about when to teach category theory. Although I try to avoid such discussion questions on MO, after some consideration I decided to add my answer. This is probably something that I'll polish a little as I think more about it, so I'm copying it here as a starting point.
I wasn't going to weigh in on this as I think that this is very definitely "subjective and argumentative" (particularly the later), and when before I spoke up in favour of category theory in undergraduate education, it sparked a few comments and I was reminded of why I like the fact that discussion is suppressed in MO. But given that one side of the argument is already here, and the other is not so well represented, I'm going to answer.
Let me start by declaring: "I am not a category theorist". I am a differential topologist. Foundational questions leave me cold, size issues just don't bother me. I'll accept any axiomatic framework if someone wants me to (I'm a fully-paid-up member of the "Axiom of Choice" party). To enter Greg's culinary world for a moment, such things are bit like Norwegian cheese. I can see that to the right person, it's delicious. But I'm not that person.
To continue the analogy, category theory isn't an ingredient that can be added for Extra Flavour, but which not everyone likes. Category theory is like cooking with freshly harvested, organic ingredients as opposed to dull, insipid, shrink-wrapped stuff from the vast conglomerate supermarket. Just making one ingredient organic doesn't have much effect on the flavour of the whole dish, but changing the whole lot does.
But to the matter in hand: undergraduates and category theory. I believe that category theory is an excellent way to understand and express mathematical concepts. I find in my own work that, time and time again, when I express my ideas using categorical language then it makes them clearer both to me and to others. Believing this, as I do, why on earth would I want to deprive my students of the same benefits?
So I teach my students category theory. I don't necessarily tell them that I'm teaching them category theory, any more than I tell them that I'm teaching them logic, or how to write proofs, or even the basics of English grammar! But I use the insights and expressions of category theory because I think it makes it easier for the students to learn "other" mathematics.
In particular, in my current course, I am trying to teach my students the following things:
To sum up, category theory isn't a "bit on the side" of mathematics to be taught as an optional extra at the higher levels, alongside homological algebra, Lie theory, and whatever-it-is-those-statisticians-down-the-hall-do. It can (and should) pervade all of our teaching because it makes the learning easier. Teaching it as a separate subject itself isn't a necessarily a bad thing, but it is if that is the only way in which it is taught, and by itself it can seem very dry, abstract, and disconnected. But then teaching it by itself is a bit like teaching logic without ever once mentioning Raymond Smullyan. Indeed, the comparison with logic is apt: we expect our students to pick up the basics of logic as they go along. Not many students actually study logic as a subject by itself, but if someone asked "Should we use logic when teaching undergraduates?" it would be closed instantly as "Not a real question.".