Why I think undergraduates will need category theory
In the comments to my first category theory post, Peter had something interesting to say. So interesting, in fact, that I decided to promote my response to an actual post.
I said “algebra” rather than “mathematics” in that post for two reasons.
First of all, yes it is a form of algebra, and the theory of small categories is every bit as useful as the theory of groups and rings and such. Categories are “like monoids but more so”, and linear categories are “like rings but more so”. Small categories are great algebraic structures in their own right, but more on that later. As for the metamathematical side, just because other forms of mathematics find such a beautiful expression in an algebraic language doesn’t make the language any less algebra.
Secondly, algebra is already seen in some circles as being further removed from reality than, say, analysis. There does exist the view that analysis is real mathematics, algebra a convenient abstraction for generalizing and streamlining analytic theories, and category theory just abstract nonsense.
As for your comparisons.. what’s so surprising about Brauer groups or Ricci flow (I must confess I’m not familiar with the Kakeya problem)? Bruce Kleiner explained Ricci flow to me in no more than five minutes, and other than all that analytic messiness Perelman worked out it seems perfectly natural that it should work. Brauer groups are even simpler, at least from the categorical viewpoint, although I’m still mystified why people want to work with the decategorification rather than the real thing.
My example of the natural transformation from the identity functor to the double-dual was just that — an example. And I brought it out because that was what gripped me when I first saw it. It’s fascinating (to me) that there’s a language and a definition of what we rhetorically call “natural”.
And why is there this undercurrent (not just in your comment) that mathematics has to be so hard to be interesting? The verification of many fine theorems is perfectly simple once you’ve set them up correctly. Grothendieck himself said he’d rather let a nut open on its own under the influence of the sun and the rain than to go at it with hammer and chisel.
What makes Yoneda’s Lemma so fascinating is not that it’s amazingly difficult, but that it’s both simple and transcendent. It’s what makes Cayley’s theorem go. It’s what makes quite a lot of modern algebraic geometry go. It’s even what makes tensor products go, which I haven’t seen written down, but I’m certain someone else had it figured out long before I stumbled upon it a couple years ago.
Now eventually I’ll be able to get right down and talk more directly about my own work, and that may satisfy you. In fact, it goes back to my first response: tangles form a category, and this allows a directly algebraic attack on knot theory, which I think will eventually become the new foundation of the subject. In fact, knot theory only becomes more and more complex the harder you fight against looking at the category of tangles. When you lay back and enjoy it, the proofs become almost effortless and the meaning becomes clear.
Now, maybe that sort of thing doesn’t appeal to you. Maybe you explicitly (not just implicitly) think that mathematics is supposed to be difficult. But there we would have to disagree. Mathematics is getting too complicated for even people who use it (like physicists) to get in all they need to know without some form of analogizing, and analogies in mathematics are called “functors”. Thinking categorically lays bare the inherent structure of an argument.
I’m not saying that every student will need to use categories in the future. I’m saying that undergraduates being conversant with basic categorical language will become pedagogically necessary in order to get through all the other mathematics they should be learning.
I first blundered into categories via physics (or, rather, by reading This Week’s Finds far too late at night). So, I’m “naturally” curious: what, in your view, are the places where basic categorical language might help physics undergraduates in their studies?
Presuming, of course, that they aren’t just double-majoring in math.
Physics undergraduates would benefit from the streamlining effect, possibly even more than mathematics undergraduates. I’ve known a few in my time, and they seem to be doing more and more calculations with less and less comprehension, largely because physics itself is getting more and more abstract. What used to be the shining city of analysis on the hill now has algebraic barbarians at the gate.
So how can they pick up even more math without sacrificing too much physics? learn overarching mathematical principles that tie field after field together, reinforcing each mathematical topic with the others.
I think of a pure mathematician as one who develops mathematics solely for the ideas involved. The problem is that such people tend to wander around in every conceivable direction, and physicists and applied math folk like myself have no idea what ideas being developed might actually be useful in their fields.
So from a somewhat selfish “outsider” perspective, I have no interest in a mathematician who says “Learn this, you will find it useful!”, and then proceeds with definition after definition and lots of proofs. I want to know at the very start how this will be useful to me. In fact I want the whole theory to be presented from the perspective of how it will be useful to me. Of course this is unrealistic, but if you are selling something to people who don’t think they need it, you need to first convince them they need it.
I remember taking a semester in differential geometry, and when we asked the instuctor how this all was used in general relativity he said he didn’t know any general relativity and had zero interest in the subject. This was a real turn off of differential geometry for those of us math/physics types.
You are right, they need it already.
There was some move to do this before but there was (and is) a lot of resistance (not from the undergraduates).
Thinking back on it, my own undergraduate education seems almost paradoxical. We wasted an absurd amount of time — basically all of sophomore year, if you followed the standard curriculum — on “introductory” classes. The rationale seemed to be, “No, the kids can’t understand linear algebra or Dirac notation, so they need to spend a semester ‘building intuition’ by solving the Schroedinger Equation over and over. Yeah, you can stick Bell’s Inequality into the last lecture, if you haven’t run out of time.”
Or, “Lagrangian mechanics is too hard, too mathematically sophisticated for sophomores, so take it out of the relativity class. Oh, that leaves an empty space? Well, fill it with toy models of black holes and cosmology. You can say they’re derived from general relativity, but better make sure they don’t require any math harder than a square root.”
Yes, the semester of solving the 1D, time-independent Schroedinger Equation was required for a physics degree. Well, in a Brazil-esque twist, it wasn’t required per se; it was merely the obligatory prerequisite to the class next in sequence, which was formally required. Why the catalog-meisters consider that a meaningful distinction, I don’t know. (Incidentally, all of the materials science and electrical-engineering types I knew who decided to take some quantum and jumped into the second term without taking the first had no trouble at all.)
But then, if you survived all that, everyone’s attitude seemed to change, and the professors were willing to try crazy things, like solving the hydrogen atom via SUSY QM or teaching a string theory class for undergraduates. I wonder if this reflects a difference between what individual professors are willing to try within a class and what the formal administration can conceive of doing for the overall curriculum.
Having read your post in response to my last comment, I
might as well comment on it: why not? The point of my previous message was to say that as someone who is in a position to choose (along with others) the undergraduate mathematical curriculum at my institution, I feel a less pressing desire to ensure a place for category theory in tha curriculum than most other traditional subjects. One must keep in mind that many (most?) undergraduates have trouble even with the “standard” level of abstraction that we try to introduce in algebra and topology — i.e., the Bourbakistic perspective of mathematics as the study of sets with additional structures. The higher level of abstraction at which category theory begins is probably too abstract for most undergraduate majors at UGA (which is at the border between a tier I and tier II institution). Having clarified this, let me respond to what you say.
It is still not clear to me that category theory is or should be a branch of algebra, but such taxonomic questions do not seem very interesting. When you say that “the theory of small categories is every bit as useful as the theory of groups and rings”, I don’t really understand how to take this statement except as a declaration of faith. Obviously there are vastly more courses, books, papers, and research specialists in group and ring theory than in small category theory, so by any sober measure the former is more useful to more mathematicians.
I don’t know what it could mean for something to be “like a ring but more so.” In mathematics certain things are interesting in themselves (and in their relations to other
specific things); the most general object is not necessarily the most interesting, and in practice overgeneralization can become boring. To tell someone that if they like some particular area of algebra (e.g. division algebras and Brauer groups) then they will love linear categories is a bit like telling a newlywed that if they like their spouse they will love all of humankind.
I won’t address the comment of whether algebra is further removed from “reality” than analysis: I don’t agree with that comment but moreover I’m not sure that it makes sense (let us not inject “reality” into a mathematical, or even meta-mathematical, discussion!).
In mentioning Brauer groups, Ricci flow and the Kakeya problem, my intention was only to mention one relatively specific subfield of algebra, geometry/topology and analysis. These are all fields which have engaged leading mathematical minds for many years, and there are many intriguing problems in each of these areas. I don’t know what to say to the comment that someone explained Ricci flow to you in five minutes: surely you don’t think that you understand “everything” about Ricci flow after this explanation?
As for Brauer groups, the issue is not whether something is “simple” or “complex” (this seems mostly a matter of perspective), but the point again is that there are many subtle problems in this area and there has been gradual and sometimes dramatic progress. It is not really practical for me to discuss such things in this format, but e.g. there is a relatively recent theorem of Johan de Jong asserting the equality of period and index in the Brauer group of the function field of a complex surface, which has set off a flurry of work. One student of de Jong has some wonderful recent results involving “twisted sheaves”, which I mention because of the nontrivial use of algebraic stacks and thus some categorical language. My point is thus not that categories are not useful, but that thinking about everything in terms of categories is not the way to go (thinking about everything in terms of _any one concept_ is not the way to go).
What is your categorical perspective on Brauer groups, and why do you think it is more useful than any of the classical perspectives? (This is not a rhetorical question.)
Mathematics is not more interesting because it is more difficult. On the other hand, you will probably agree that if every theorem in mathematics were straightforward to prove than the subject _would_ become less interesting. One of the charms of the subject is that it is not clear how hard one may have to work to establish something (if in fact one can establish it at all). All things being equal, an interesting theorem with a nice, easy proof is muh to be preferred to an interesting theorem with a proof that takes the experts years to understand. Congratulations to Yoneda for discovering his lemma: it’s great. But it is not always the case that things work this way. Grothendieck’s skill in finding the apparently fluffily general theory that turns out to be perfectly adapted to the problem at hand is amazing and unparalleled (I admire him very much), but even he could not solve all problems by marination. The famous example is the Weil conjectures, two-thirds of which were cracked by finding exactly the right perspective (l-adic cohomology) but the last and greatest third required Deligne’s more “concrete” contribution. The interaction of relatively abstract and relatively concrete is characteristic of modern mathematical research — it’s no good to keep your head in the clouds all the time.
When you say that knowledge of some category theory will become pedgagogically necessary at some point for reasons of data compression…I agree, and I think we are there now. Starting at the basic graduate level I agree that one should call an adjoint functor an adjoint functor and that this makes many things quicker and easier to understand. But, once again, I can’t picture a graduate course in category theory per se but rather a systematic use of very basic category-theoretic language in all courses in much the same way that very basic set theory has been assumed and used for many years now.
And how much older are those subjects?
I may have misinterpreted. You seemed to be saying that categories were inherently not as “deep” or “interesting” as those fields. No, I don’t claim to be a master of any of them, but I do have a certain understanding of the general outlines of the ones I am familiar with, and the general outlines are simple. The devil — along with the depth — is in the details. You’re asking me to compare the general outlines of category theory with the fine details of those three problems, and it’s not a fair comparison.
Okay, here’s an example of the details: In their work on categorifying holonomies, Baez and Schreiber showed how the categorical viewpoint makes natural a “fake flatness” condition that physicists had been using as an ansatz because it seemed to make their lives easier. Or how about Jeffrey Morton’s categorified version of the quantum harmonic oscillator that seems to produce Feynman diagrams out of thin air? And did I hear someone mentioning Khovanov’s solution of the Milnor conjecture?
No, I’m not saying that we should teach the whole of category theory to undergraduates, but neither do we teach more than a thumbnail sketch of any other field of mathematics at that level.
As for your specific question on Brauer groups, I’ll get there eventually. The basics can be found if you search in This Week’s Finds for the term.
It’s a little subtler than that. No, it wouldn’t be interesting if it were completely obvious, but what’s most interesting to me is that fact that so many areas are simple when you look at them just the right way. The way to see a given problem may be hard to find, but once you find it everything falls into place so naturally it’s hard to imagine that it could be another way. Hand me a theorem in analysis that takes three pages of estimates and bounds to work through and my eyes glaze over. I’m glad someone does it, yes, but I’m more glad that that someone isn’t me.
Another place we seem to be talking past each other is in the timescale. I say “soon”, meaning “the next few decades”. For now the benefit is in introducing the unifying themes, whether we say the word “category” or not. For instance: teach group theory the same as ever, but mention the universal properties of direct and free products.
My statement that category theory qua category theory will filter down to the undergraduate level in the coming decades (not years) is a prediction, and prediction is difficult — especially about the future. I may well be wrong, but my prediction isn’t without precedent. Scott Carter quoted Herstein’s Abstract Algebra in the comment I linked to recently, and it’s apropos in this context as well:
I look back, and I look forward, and I predict categories will follow the older parts of abstract algebra into the undergraduate curriculum.
“And how much older are those subjects?”
That is hard to say. But it is disingenuous of you
to imply that category theory is such a new field that its proportion of the mathematical landscape cannot be accurately estimated. Category theory was created by Eilenberg and Mac Lane in the early 1940’s (so it predates Galois cohomology and far predates Ricci flow).
I just looked at the list of Graduate Texts in Mathematics
(on wikipedia) and found that exactly one out of 242 is devoted to category theory: Mac Lane’s “Categories for the working mathematician”, from 1971. By comparison, about 20-25% of the GTM’s are devoted to conventional algebraic
topics. I don’t necessarily approve of this less than one
percent representation, but those are the facts (and similarly, we could certainly verify that self-identified
category theorists make up less than one percent of math department faculty). You need to distinguish what is true
now from what you hope will be true in the future.
Yes, I am asking you to compare things which are more general with things which are more specific, and to notice that apples are being compared to oranges. Category _theory_ operates in more generality than most other disciplines of conventional mathematics, and my position
(and not just mine, of course) is that this generality, while it can be very useful in applications, is comparatively uninteresting in and of itself.
I should interject that I know nothing about n-categories for n > 1 (the set-theoretical difficulties involved make me a little nervous, but more honestly, my own research has thus far not taken me down that path). I can only say that to the extent that this higher category theory is becoming useful in contemporary mathematics, it seems to have a different flavor from “classical” category theory and one which is much more reminiscent of conventional mathematics.
(I don’t know what you heard, but Khovanov did not prove the Milnor conjecture. Rasmussen gave a second proof using Khovanov homology.)
It is nice to find a solution to a difficult problem which is, in retrospect, natural and obvious — very nice! I am not an analyst either, and I find it strange that so many analysis talks consist of a flurry of detailed calculations that seem to be best dealt with in the privacy of one’s own home or office. But not everything gets a limpid, “purely conceptual” proof, even in retrospect. I already gave you an example of that: the Weil Conjectures. Or still more: the classification of finite simple groups. And (of course, right?) there are other routes to conceptualization besides just categorifying everything in sight.
Gian-Carlo Rota was of the opinion that category theorists have been making a PR mistake by pretending to work towards the recasting of all mathematics in the language of category theory. Most mathematicians have a distaste for foundationalism and do not like being told that the objects of their affection need to be conceived of and spoken about via some particular machinery imposed from afar. Since what is puzzling about category theory is how little a subject that most mathematicians agree has something useful to offer has filtered down to the standard under/graduate vocabulary, maybe you should think about how the PR can be improved. (Having a blog that explains basic concepts is a really good idea!)
Your quote from Herstein is about 40 years old. Probably the filtration of knowledge down to the undergraduate curriculum did continue into the 70’s and beyond. I am here to tell you now that what you and I agree should be in the undergraduate curriculum (and everything we have both said so far indicates that we are in near-identical agreement on what the undergraduate curriculum _should_ be) is in serious danger of being removed from the curriculum at many — even rather good — institutions. Herstein’s excellent book (which I used in my undergraduate algebra couse) played a large role in fulfilling his prediction. We should think similarly about how to ensure that our future hopes and predictions will come to pass.
First, I must apologize for my misstatement regarding the Milnor conjecture. I should know better than to mix mathematics with packing at 01:30.
Now, I’m not saying that category theory is in its infancy, but it’s certainly not as old as other fields, and directly practical (as opposed to metamathematical) applications are newer still. However, it’s not really reflective of its importance to count book titles. There may be few directly and solely on categories, but a good portion of of algebraic geometry, algebraic topology, representation theory, mathematical logic, or sundry other fields uses the language of categories, functors, and natural transformations.
It seems that you’re reading my use of “category theory” to mean categories for their own sake, which simply isn’t the case. We teach group theory not only to prepare students to study groups, but also to use groups in studying all sorts of other areas of mathematics. The same will be true (I believe) of categories.
I’m also not advocating teaching categories as a new foundation, though personally I think it may be a (much longer-term) possibility to use them as such.
And yes, other areas of mathematics are jeopardized in the undergraduate curriculum. This is another, separate, problem that also needs to be dealt with. I don’t think that category theory needs to displace other subjects, though.
I represent a typical “interested outsider” because I don’t have much of a formal mathematical education. I have a degree from economics, 15 years of working experiences with all kinds of investments and now I’m working on my Phd thesis in Artificial Intelligence ( http://viljem.tisnikar.googlepages.com/home ).
What I wanted to share with you is the experience I had with Category Theory (CT). Two years back from now I did not know even of existence of such a theory. Some years ago I designed some innovative solutions for software of type “computer supported cooperative work”. These solutions boosted our productivity enormously and I felt that we found something really important.
I’ve begun to search for answers, tried to build a coherent mathematical model, spoken with fine mathematicians (mostly working on graph theory) but step by step Google hits brought to me more and more articles containing “abstract nonsense”. For all my intuitive ideas and findings from practice I found some construction in CT! Now I am working (for my Phd thesis) on CT for Knowledge Management, an entirely new field up to my knowledge.
For George Bell and others let me show the usefulness of CT with two links to other people’s work of quite different nature: http://www.ellerman.org/Davids-Stuff/AboutDavidEllerman.htm and http://www.worldscibooks.com/mathematics/5968.html
Ellerman uses CT as a tool for explaining complex socio-economical relations in the world and proposes how to improve the world for all humanity. What more do you need?
Ivančevič models human biodynamics with heavy math artillery but for putting all together and for modeling brain-body relation they need CT (adjunction).
Yes, I agree that undergraduates need CT but I believe a lot of work should be done with literature. I’m (self) learning CT for a year and half now and I like it. But obviously it requires a certain profile of people who are able to learn CT from existing literature. I believe that such foundational and general things CT is dealing with could be represented better. MacLane is a poetry but not everyone understands poetry. Awodey made a good step in the “not only for the working mathematician” direction but it isn’t enough. Some parts of his book are brilliant but some are bad (from pedagogical point of view, of course).
I’ve seen category theory taught in two different settings. I myself was taught category theory at the end of an undergraduate maths degree. I now work in computer science and category theory here is taught to grad students.
The problem with the CS course is that providing accessible examples to students with a CS background is difficult. You can’t use topology or even too much algebra; even group theory sends some of them running. It was wonderful and enlightening to see a whole pile of things from my undergrad course turn out to be examples of adjunctions (say) but the CS students don’t know any of that pile of things to start with.
I used to think that undergrads should know CT early on, but now I believe that it’s very hard to motivate such generality before providing more specific examples that have a common theme at the CT level.
David, you make a good point. I’ll say, though, that I’m mainly talking about mathematics undergraduates, and I’ve only recently come across departments in which one can get a bachelor’s degree in mathematics without taking an abstract algebra course.
Besides, what you say is why I ran through groups, rings, and modules before starting in on categories.
I’ve started Wikiversity learning project on Category Theory at undergraduate level. All help/feedback is appreciated. Just click my name.