My 2019 Mathematics A To Z: Category Theory


Today’s A To Z term is category theory. It was suggested by aajohannas, on Twitter as @aajohannas. It’s a topic I have long wanted to know better, and that every year or so I make a new attempt to try learning without ever feeling like I’ve made progress.

The language of it is beautiful, though. Much of its work is attractive just to see, too, as the field’s developed notation that could be presented as visual art. Much of mathematics could be visual art, yes, but these are art you can almost create in ASCII. It’s amazing.

Cartoony banner illustration of a coati, a raccoon-like animal, flying a kite in the clear autumn sky. A skywriting plane has written 'MATHEMATIC A TO Z'; the kite, with the letter 'S' on it to make the word 'MATHEMATICS'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Category Theory.

What is the most important part of mathematics? Well, the part you wish you understood, yes. But what’s the fundamental part? The piece of mathematics that we could feel most sure an alien intelligence would agree is mathematics?

There’s idle curiosity behind this, yes. It’s a question implicit in some ideals of the Enlightenment. The notion that we should be able to find truths that all beings capable of reason would agree upon, and find themselves. Mathematics seems particularly good for this. If we have something proven by deductive logic from clearly stated axioms and definitions, then we know something true.

There’s practicality too. In the late 19th and early 20th century (western) mathematics tried to find logically rigorous foundations. You might have thought we always had that and, uh, not so much. It turns out a complete rigorous logical proof of even simple stuff takes forever. Mathematicians compose enough of an argument to convince other mathematicians that we could fill in the details. But we still trusted there was a rigorous foundation. The question is, what is it?

A great candidate for this was set theory. This was a great breakthrough. The basic modern idea of set theory builds on bunches of things, called elements. And collections of those things, called sets. And we build rigorous ideas of what it means for elements to be members of sets. This doesn’t sound like much. Powerful ideas never do.

I don’t know that everyone’s intuition is like this. But my gut wants to think a “powerful” result is, like, a great rocket. Some enormous and prominent and mighty thing that blasts through a problem like gravity or an atmosphere. This is almost the opposite of what mathematics means by “powerful”. A rocket is a fiddly, delicate thing. It has millions of components made to tight specifications. It can only launch when lots of conditions are exactly right. A theorem that gives a great result, but has a long list of prerequisites and lemmas that feed into it resembles this. A powerful mathematical result is more like the gravity that the rocket overcomes. It tends to suppose little about the situation, and so it provides results that are applicable over the whole field. Or over a wide field, or a surprising breadth of topics. And, really, mighty as a rocket might be, the gravity it fights is moreso.

So set theory is powerful. It can explain many things. Most amazing is that we can represent arithmetic with it. At least we can get to integers, and all that we do with integers, and that in not too much work. It makes sense that mathematicians latched onto this as critical. It fueled much of the thinking behind the New Math, the infamous attempted United States educational reform of the 1960s and 70s. I grew up in the tail end of this, learning unions and intersections and complements along with times tables and delighted in it.

But even before New Math became a coherent idea there was a better idea. Emmy Noether, mentioned yesterday, is not a part of it. But a part of her insight into physics, and into group theory, was an understanding of structure. That important mathematics results from considering what we can do with sets of things. And what things we can do that produce invariants, things that don’t change. Saunders Mac Lane, one of Noether’s students, and Samuel Eilenberg in the 1940s used what looks to me like this principle. They organized category theory.

Category Theory looks at first like set theory only made terrifying. I’m not very comfortable with it myself. It’s an abstract field, and I’m more at home with stuff I can write a quick Octave program to double-check. Many results in category theory are described, or even proved, with beautiful directed-graph lattices. They show how things relate to one another. This is definitely the field to study if you like drawing arrows.

Part of the statement of a theorem, which includes three squares showing relations between objects labelled A and A', D(I) and D'(I), lim D and lim D'.
I admit I don’t know what point is being proved here. But the square structure illustrated here, and the way successive squares will, say, replace lim D with A, or pI with fI, is almost a sestina. From page 145 of Dr Tom Leinster’s Basic Category Theory.

Just as set theory does, category theory starts with things, called objects. And these objects get piled together into collections. And then there’s another collection of relationships between these collections. These relationships you call maps or morphisms or arrows, based on whatever the first book you kind of understood called them. I’m partial to “maps”. And then we have rules by which these maps compose, that is, where two maps reduce to a single map. This bundle of things — the objects, the collections, and the maps — is a category.

These objects can start out looking like elements, and the collections like sets, and the maps like functions. This gives me, at least, a patch of ground where I feel like I know what I’m doing. But what we need of things to be objects and collections and maps is very little. The result is great power. We can describe set theory in the language of categories. So we can describe arithmetic in category theory. There’s a bit of a hike from the start of category theory to, like, knowing what 18 plus 7 is.

But we’re not bound to anything that concrete. We can describe, for example, groups as categories. This gives us results like when we can factor polynomials. Or whether compass and straightedge can trisect an arbitrary angle. (There’s some work behind this too.) We can describe vector spaces as categories. Heady results like the idea that one function might be orthogonal to another lurk within this field. Manifolds, spaces that work like normal space, are part of the field. So are topological spaces, which tell us about shapes.

If you aren’t yet dizzy then consider this. A category is itself an object. So we can define maps between categories. These we call functors. Which themselves have use in computer science, as a way some kinds of software can be programmed well. More, maps themselves are objects. We can define mappings between maps. These we call natural transformations. Which are the things that Eilenberg and Mac Lane were particularly interested in, to start with. Category theory grew in part out of needing a better understanding of natural transformations.

I do not know what to recommend for people who want to really learn category theory. I haven’t found the textbook or the blog that makes me feel like I am mastering the subject. Writing this essay has introduced me to Dr Tom Leinster’s Basic Category Theory, which I’ve enjoyed skimming. Exercise 3.3.1, for example, seems like exactly the sort of problem I would pose if I knew category theory well enough to write a book on it.

Is this, finally, the mathematics we could be sure an alien would recognize? I’m skeptical, but I always am. It seems to me we build mathematics on arithmetic and geometry. Category theory, seeming to offer explanations of both, is a natural foundation for that. But we are evolved to see the world in terms of number and shape. Of course we see arithmetic and geometry as mathematics. Can we count on every being capable of reason seeing the same things as important? … I admit I can’t imagine a being we might communicate with not recognizing both. But this may say more about the limits of my imagination than about the limits of what could be mathematics.


Thanks for reading. All the Fall 2019 A To Z posts should be at this link. I hope to have the second essay of the week posted Thursday. This year’s and all past A To Z sequences should be at this link. And if you have thoughts about other topics I might cover, please offer suggestions for the letters F through H.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

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