I Served Serial Joe a Slushie

The State of Category Theory

Recently, I heard an interesting story from a student. In his math class (applied math of some kind) the teacher had a Tim Horton's cup. He rolled up the rim, looked it, said "Oh, a PhD in category theory" and tossed it in the garbage. Cue big laughs.

Being the only category Theory graduate student at the time (there are now two others here), hearing this hurt. I'm always worried about job prospects, and having people denounce them to all around makes it worse - if people's opinions are like this, what chance do I have to get hired? Thinking about this forced me to reconsider my own opinions about category theory.

When I first heard of category theory, it seemed slightly useless to me. Instead of considering, say, groups, you consider the "category" of all groups? What's the point? Perhaps, say, it becomes easier to understand products of groups and such, but still, it didn't seem that useful. It was only once I took my class here that I realized that a big part of the attraction was that one could consider not just the category of all groups, but a group is itself a type of category: one with only one object, and each morphism having an inverse. Continuing on this came the insight that an ordered set is also itself a type of category: one with at most one morphism between any two objects. Later, I learned that metric spaces themselves could also be viewed as certain types of "enriched" categories.

One is then led to a fascinating interaction between the very small categories which are themselves groups, or ordered sets, or metric spaces, and the very large categories of all groups, or all metric spaces, and so on. For example, a functor from a group (considered as a category )to the category of all k-linear spaces is exactly a representation of that group. I started to become convinced of the incredible power of these ideas. In my heart I felt category theory was "right"; it didn't matter that other people hadn't seen this yet; only time was needed to saturate the ideas throughout the mathematical community.

Of course, this hardly helps with job prospects, and so even though I myself was convinced of it's usefulness, that hardly mattered when I wanted to get hired. But lately I've started to realize something: how pervasive categorical ideas already are; I just wasn't aware of the examples until recently.

My first example came last spring when I was taking an algebraic topology course. The instructor did not use any category theory, but once I started looking in books, I realized that nearly every book out there did - homology theories as functors, limits of topological spaces, it was all there.

My second example came in the category theory seminar, when I realized that the theory of "cartesian closed categories" was exactly the same as the lambda calculus. When I investigated a bit further, I found dozens of message board with computer scientists eager to learn about adjoint functors, monads, and other categorical ideas, to describe their computational ideas.

The third example came recently. Through a rather side-ways route, I learned about "schemes" a Grothendieck idea to help discuss algebraic geometry. I looked at it, thought "it's category theory, it probably only has a small following" - only to discover that "some consider schemes to be the basic object of study of modern algebraic geometry". When I asked a classmate about schemes who had recently taken a course in algebraic geometry, he said "oh ya, we did it in the last half of the course - it was a lot easier to understand than the classical stuff".

Perhaps I needn't worry so much.