Andrew Stacey

Andrew Stacey
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By: Andrew Stacey
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Andrew Stacey

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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:

  1. To focus on processes rather than things. Call them "morphisms" and "objects" and that's category theory! I don't tell them to do this because that's what category theory tells us to do, I tell them to do this because that's what the Real World(TM) tells us to do: mathematics (I tell them) is about modelling the real world, and the basic thing that one wants to model is a process.

  2. To transfer knowledge from a known space to an unknown space. Here we have the extension of the mathematical idea of "function over form". That is, a thing is not defined by what it is (object) but what it does (what category it is in). But we can take this one step further and say that it's not just what it itself does that matters, but how it relates to the things around it (what morphisms are there from it to other objects in the category?). In particular, if I have an unknown vector space $V$ (unknown in the sense that I don't know much about it rather than I don't know how to define it), I gain a lot of knowledge if I can find an isomorphism $V \cong \mathbb{R}^n$ because I already know a lot about $\mathbb{R}^n$.

    In a recent colloquium, I made this point (rather strongly) by saying that category theory is ubuntu mathematics: "I am what I am because of who we all are."

  3. To be able to change ones point of view to suit the problem at hand. Say, "to look for what is preserved under isomorphism" and you've got one of the central tenets of category theory: that isomorphic objects should not be distinguished. This is a natural extension of the above. Once we know that an isomorphism $V \cong \mathbb{R}^n$ is a Good Thing, the next question is whether or not there's a best isomorphism (for the problem at hand).

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.".

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Thu, 7th Oct 2010