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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The Summer 2017 Mathematics A To Z: Functor


Gaurish gives me another topic for today. I’m now no longer sure whether Gaurish hopes me to become a topology blogger or a category theory blogger. I have the last laugh, though. I’ve wanted to get better-versed in both fields and there’s nothing like explaining something to learn about it.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Functor.

So, category theory. It’s a foundational field. It talks about stuff that’s terribly abstract. This means it’s powerful, but it can be hard to think of interesting examples. I’ll try, though.

It starts with categories. These have three parts. The first part is a set of things. (There always is.) The second part is a collection of matches between pairs of things in the set. They’re called morphisms. The third part is a rule that lets us combine two morphisms into a new, third one. That is. Suppose ‘a’, ‘b’, and ‘c’ are things in the set. Then there’s a morphism that matches a \rightarrow b , and a morphism that matches b \rightarrow c . And we can combine them into another morphism that matches a \rightarrow c . So we have a set of things, and a set of things we can do with those things. And the set of things we can do is itself a group.

This describes a lot of stuff. Group theory fits seamlessly into this description. Most of what we do with numbers is a kind of group theory. Vector spaces do too. Most of what we do with analysis has vector spaces underneath it. Topology does too. Most of what we do with geometry is an expression of topology. So you see why category theory is so foundational.

Functors enter our picture when we have two categories. Or more. They’re about the ways we can match up categories. But let’s start with two categories. One of them I’ll name ‘C’, and the other, ‘D’. A functor has to match everything that’s in the set of ‘C’ to something that’s in the set of ‘D’.

And it does more. It has to match every morphism between things in ‘C’ to some other morphism, between corresponding things in ‘D’. It’s got to do it in a way that satisfies that combining, too. That is, suppose that ‘f’ and ‘g’ are morphisms for ‘C’. And that ‘f’ and ‘g’ combine to make ‘h’. Then, the functor has to match ‘f’ and ‘g’ and ‘h’ to some morphisms for ‘D’. The combination of whatever ‘f’ matches to and whatever ‘g’ matches to has to be whatever ‘h’ matches to.

This might sound to you like a homomorphism. If it does, I admire your memory or mathematical prowess. Functors are about matching one thing to another in a way that preserves structure. Structure is the way that sets of things can interact. We naturally look for stuff made up of different things that have the same structure. Yes, functors are themselves a category. That is, you can make a brand-new category whose set of things are the functors between two other categories. This is a good spot to pause while the dizziness passes.

There are two kingdoms of functor. You tell them apart by what they do with the morphisms. Here again I’m going to need my categories ‘C’ and ‘D’. I need a morphism for ‘C’. I’ll call that ‘f’. ‘f’ has to match something in the set of ‘C’ to something in the set of ‘C’. Let me call the first something ‘a’, and the second something ‘b’. That’s all right so far? Thank you.

Let me call my functor ‘F’. ‘F’ matches all the elements in ‘C’ to elements in ‘D’. And it matches all the morphisms on the elements in ‘C’ to morphisms on the elmenets in ‘D’. So if I write ‘F(a)’, what I mean is look at the element ‘a’ in the set for ‘C’. Then look at what element in the set for ‘D’ the functor matches with ‘a’. If I write ‘F(b)’, what I mean is look at the element ‘b’ in the set for ‘C’. Then pick out whatever element in the set for ‘D’ gets matched to ‘b’. If I write ‘F(f)’, what I mean is to look at the morphism ‘f’ between elements in ‘C’. Then pick out whatever morphism between elements in ‘D’ that that gets matched with.

Here’s where I’m going with this. Suppose my morphism ‘f’ matches ‘a’ to ‘b’. Does the functor of that morphism, ‘F(f)’, match ‘F(a)’ to ‘F(b)’? Of course, you say, what else could it do? And the answer is: why couldn’t it match ‘F(b)’ to ‘F(a)’?

No, it doesn’t break everything. Not if you’re consistent about swapping the order of the matchings. The normal everyday order, the one you’d thought couldn’t have an alternative, is a “covariant functor”. The crosswise order, this second thought, is a “contravariant functor”. Covariant and contravariant are distinctions that weave through much of mathematics. They particularly appear through tensors and the geometry they imply. In that introduction they tend to be difficult, even mean, creations, since in regular old Euclidean space they don’t mean anything different. They’re different for non-Euclidean spaces, and that’s important and valuable. The covariant versus contravariant difference is easier to grasp here.

Functors work their way into computer science. The avenue here is in functional programming. That’s a method of programming in which instead of the normal long list of commands, you write a single line of code that holds like fourteen “->” symbols that makes the computer stop and catch fire when it encounters a bug. The advantage is that when you have the code debugged it’s quite speedy and memory-efficient. The disadvantage is if you have to alter the function later, it’s easiest to throw everything out and start from scratch, beginning from vacuum-tube-based computing machines. But it works well while it does. You just have to get the hang of it.

The Summer 2017 Mathematics A To Z: Cohomology


Today’s A To Z topic is another request from Gaurish, of the For The Love Of Mathematics blog. Also part of what looks like a quest to make me become a topology blogger, at least for a little while. It’s going to be exciting and I hope not to faceplant as I try this.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Also, a note about Thomas K Dye, who’s drawn the banner art for this and for the Why Stuff Can Orbit series: the publisher for collections of his comic strip is having a sale this weekend.

Cohomology.

The word looks intimidating, and faintly of technobabble. It’s less cryptic than it appears. We see parts of it in non-mathematical contexts. In biology class we would see “homology”, the sharing of structure in body parts that look superficially very different. We also see it in art class. The instructor points out that a dog’s leg looks like that because they stand on their toes. What looks like a backward-facing knee is just the ankle, and if we stand on our toes we see that in ourselves. We might see it in chemistry, as many interesting organic compounds differ only in how long or how numerous the boring parts are. The stuff that does work is the same, or close to the same. And this is a hint to what a mathematician means by cohomology. It’s something in shapes. It’s particularly something in how different things might have similar shapes. Yes, I am using a homology in language here.

I often talk casually about the “shape” of mathematical things. Or their “structures”. This sounds weird and abstract to start and never really gets better. We can get some footing if we think about drawing the thing we’re talking about. Could we represent the thing we’re working on as a figure? Often we can. Maybe we can draw a polygon, with the vertices of the shape matching the pieces of our mathematical thing. We get the structure of our thing from thinking about what we can do to that polygon without changing the way it looks. Or without changing the way we can do whatever our original mathematical thing does.

This leads us to homologies. We get them by looking for stuff that’s true even if we moosh up the original thing. The classic homology comes from polyhedrons, three-dimensional shapes. There’s a relationship between the number of vertices, the number of edges, and the number of faces of a polyhedron. It doesn’t change even if you stretch the shape out longer, or squish it down, for that matter slice off a corner. It only changes if you punch a new hole through the middle of it. Or if you plug one up. That would be unsporting. A homology describes something about the structure of a mathematical thing. It might even be literal. Topology, the study of what we know about shapes without bringing distance into it, has the number of holes that go through a thing as a homology. This gets feeling like a comfortable, familiar idea now.

But that isn’t a cohomology. That ‘co’ prefix looks dangerous. At least it looks significant. When the ‘co’ prefix has turned up before it’s meant something is shaped by how it refers to something else. Coordinates aren’t just number lines; they’re collections of number lines that we can use to say where things are. If ‘a’ is a factor of the number ‘x’, its cofactor is the number you multiply ‘a’ by in order to get ‘x’. (For real numbers that’s just x divided by a. For other stuff it might be weirder.) A codomain is a set that a function maps a domain into (and must contain the range, at least). Cosets aren’t just sets; they’re ways we can divide (for example) the counting numbers into odds and evens.

So what’s the ‘co’ part for a homology? I’m sad to say we start losing that comfortable feeling now. We have to look at something we’re used to thinking of as a process as though it were a thing. These things are morphisms: what are the ways we can match one mathematical structure to another? Sometimes the morphisms are easy. We can match the even numbers up with all the integers: match 0 with 0, match 2 with 1, match -6 with -3, and so on. Addition on the even numbers matches with addition on the integers: 4 plus 6 is 10; 2 plus 3 is 5. For that matter, we can match the integers with the multiples of three: match 1 with 3, match -1 with -3, match 5 with 15. 1 plus -2 is -1; 3 plus -6 is -9.

What happens if we look at the sets of matchings that we can do as if that were a set of things? That is, not some human concept like ‘2’ but rather ‘match a number with one-half its value’? And ‘match a number with three times its value’? These can be the population of a new set of things.

And these things can interact. Suppose we “match a number with one-half its value” and then immediately “match a number with three times its value”. Can we do that? … Sure, easily. 4 matches to 2 which goes on to 6. 8 matches to 4 which goes on to 12. Can we write that as a single matching? Again, sure. 4 matches to 6. 8 matches to 12. -2 matches to -3. We can write this as “match a number with three-halves its value”. We’ve taken “match a number with one-half its value” and combined it with “match a number with three times its value”. And it’s given us the new “match a number with three-halves its value”. These things we can do to the integers are themselves things that can interact.

This is a good moment to pause and let the dizziness pass.

It isn’t just you. There is something weird thinking of “doing stuff to a set” as a thing. And we have to get a touch more abstract than even this. We should be all right, but please do not try not to use this to defend your thesis in category theory. Just use it to not look forlorn when talking to your friend who’s defending her thesis in category theory.

Now, we can take this collection of all the ways we can relate one set of things to another. And we can combine this with an operation that works kind of like addition. Some way to “add” one way-to-match-things to another and get a way-to-match-things. There’s also something that works kind of like multiplication. It’s a different way to combine these ways-to-match-things. This forms a ring, which is a kind of structure that mathematicians learn about in Introduction to Not That Kind Of Algebra. There are many constructs that are rings. The integers, for example, are also a ring, with addition and multiplication the same old processes we’ve always used.

And just as we can sort the integers into odds and evens — or into other groupings, like “multiples of three” and “one plus a multiple of three” and “two plus a multiple of three” — so we can sort the ways-to-match-things into new collections. And this is our cohomology. It’s the ways we can sort and classify the different ways to manipulate whatever we started on.

I apologize that this sounds so abstract as to barely exist. I admit we’re far from a nice solid example such as “six”. But the abstractness is what gives cohomologies explanatory power. We depend very little on the specifics of what we might talk about. And therefore what we can prove is true for very many things. It takes a while to get there, is all.