I remember back when I was in high school and came across lists of the greatest mathematicians ever. They almost always included Archimedes, Newton and Gauss. Sometimes Euler made it in. I knew who these guys were, but every once in a while, there was this guy I had never heard of, Alexander Grothendieck. I with pretty much no idea what he had done until I hit graduate school where I began to appreciate his contributions to/invention of modern mathematics. I’d like to talk a little about a philosophical aspect of his work here. I don’t know the history so well, and I’m sure all these ideas were developed by a group of people, but I think it safe to say that Grothendieck was central to this.
Before the middle of the twentieth century, mathematician studied various objects in terms of what they were made up of. To get the usual number systems, for example, Peano defines the rules that a set of objects have to obey to be the natural numbers, and from there, you can build up the integers, the rationals, and the real and complex numbers. In most algebraic structures, you have a set of objects, some operations you can perform on them, and various properties that the operations have to satisfy. A particularly simple example is called a “group”. A group roughly consists of a set of objects and rules for multiplying and dividing them. (For the experts, yes, I know. Anyone interested in the precise definition of a group can see here.) Thus, if we have two objects ‘a’ and ‘b’, in a group we have to have another object ‘ab’ which is the product of ‘a’ and ‘b’. We also require that this product is “associative” which means that
Or, in words, if we multiply ‘a’ and ‘b’ and then multiply the result by ‘c’, we get the same thing as if we had multiplied ‘b’ and ‘c’ first and then multiplied the result by ‘a’.
This is a particularly simple example of a structure you can put on a set, abstracting some of the properties of the rational numbers. You can add more and more operations and properties, and a fair chunk of mathematics is studying what you get.
For years, sets seemed such an intuitive and obvious notion that they served as a foundation for mathematics. Various axiomatics for set theory were developed, and people studied what could and could not be proven from those axioms. Much fun was had. Grothendieck’s great insight, or at least the great insight I will assign to Grothendieck because I don’t know the history so well, is that there is another way one can look at structures in mathematics. Instead of thinking about constituents and what we can do to constituents, we should instead think about how things relate to each other.
What does this mean? Let’s rephrase the notion of a group in this language. The point of a group is that we can multiply things. We can rephrase this as a function:
What are relating here? Well, on one side, we have two copies of the group, G. On the other side, we have one copy of the group. So, we have a map from two copies of the group to one copy of the group. We write this as an arrow:
G x G is just a way to say that we have two copies of the group. Recall the associativity rule:
Rewritten in terms of functions, this is
The left and right sides of this equation are maps from three copies of the group (which we write G x G x G) to G. So, we have two maps
that we form out of our m. Associativity is the statement that these two maps are equal. Mathematicians have a fancy way of writing this called a commutative diagram:
What this diagram means is that if we follow the arrows in either order, we arrive at the same thing. The symbol “id” is the identity map from G to itself. The composition of the upper and right arrows corresponds to the right side of the above equation expressing associativity, and the composition of the left and lower arrows correponds to the left side. Groups also have division and an identity (something so that ai = a and ia = a), but we’ll neglect those. If you know about them, you’ll find it easy to rephrase division (by which I mean, of course, inverses) in this language. The identity takes another bit of data (called a terminal object) which I’ll mention in a moment.
What we have done is to rephrase this notion of a group without ever talking about the constituents of it. A group is just a set with various ways of relating its self-product to itself satisfying certain properties. In fact, we only need to understand a few things to say what a group is. We need to know what an arrow and a commutative diagram are. We need to know what a product is, so that something like G x G makes sense. And lastly, we need to know what a ‘terminal object’ is.
Let’s start with the first. An arrow is a way of saying that two things relate to each other. It can be by a function or by anything else. The formalization of this idea is something called a category. Modulo some details, a category is simply a collection of objects and arrows between them. We need to say that if we travel along consecutive arrows, we can obtain another arrow. This is called composition and is the statement that if A relates to B in some manner and if B relates to C in a similar way, then A relates to C. For example, as suggested by the name, if we’re relating sets by functions between them, the composition of arrows is composition of functions. (nb: I’m using “relation” in a hopefully intuitive way here, not in any technical sense.)
So, if we have a category around, we are saying that we have a bunch of things, and we know how they can relate to each other. This is sufficient to be able to make commutative diagrams. There is an amazing amount one can say with just this simple structure. Because of its generality, most (every?) objects in mathematics can usefully be thought of as living in a category, and everything we can say about categories in general can then be applied to specific cases. In this way, one can understand many phenomena that occur in a broad array of seemingly unrelated situations. They’re all examples of category theory.
But, as fun as categories are, they’re not enough to get us a group. But we don’t need much more. The first thing we need is to be able to take products. Thus, if we have two objects A and B, we need to be able to obtain a new object A x B. I’ll chicken out of trying to explain exactly what that object needs to do to be considered a product. Finally, we need this terminal object which is an object such that there is a unique arrow to it from every other object. The identity arises as a map from this object to the group. If we have these extra structures on a category, we can define a “group object” in it. By a group object, I mean an object in the category, a map from G x G -> G obeying the above commutative diagram, and the various maps and diagrams that give the other properties of groups.
If we think of all sets and functions between them, that’s a category, so a group as we initially defined it is a group object in the category Set. But we don’t need to stop there. All categories with products and terminal objects admit the possibility of group objects. Much of group theory applies to group objects as well as ordinary groups, ie, group objects in Set. So, by phrasing things in terms of how things relate to each other, we’ve immediately found ourselves tons of new objects that we’ve already proven things about.
If you find this stuff fun, ask yourself what a terminal object is in the category of Set. Then figure out what an element of a set is in this relational language.
This might seem like a silly exercise in abstraction, but thinking this way can be immensely powerful. There’s a famous theorem in geometry called the Hirzebruch-Riemann-Roch theorem. At the risk of being so sketchy so as to be completely incomprehensible, this theorem takes as its input a space and something that lives on it, a “coherent sheaf”. Out of this, you can associate a number in two different ways. One involves something called a K-group, and the other involves something called a Chow group. There are ways of getting numbers out of each. The theorem gives a map from the K-group to the Chow group such that if we start with an object in the K-group, map it to the Chow group and get a number, it’s the same thing as just getting the number from the K-group.
What Grothendieck realized is that the both the K-theory and the Chow group of a point give numbers and that the integers obtained from them as above are associated to the map from the space to a single point. Thus, instead of being a theorem about numbers associated to spaces with sheaves on them, we should think of the theorem as being about how the space and the sheaf on it relate to the point (there is only one map from any space to a single point making it a terminal object). Once you think about the theorem this way, there’s no particular reason to only relate your space to a point. You can look at pairs of spaces with nice enough maps between them. Let’s call these spaces X and Y and the nice map f. To each space, we have the K-group which we’ll write as K(X) and K(Y) and the Chow group which we’ll write as A(X) and A(Y) (for the experts, I’m implicitly assuming rational coefficients). Then, we can take the previous map which we’ll call
and get a commutative diagram.
The horizontal rows here are the map i and the vertical lines are maps of K-groups and Chow groups that arise from the map f between the spaces X and Y. Since I’ve probably lost everyone who didn’t already know this at this point, let me just say what the moral is. What everyone had thought to be a theorem about spaces instead turned out to be a theorem about relations between spaces. Before Grothendieck, no one had realized that they were only looking at relations between spaces and a point. This perspective has lead to many other new theorems and restatements of old theorems in this relative context.
What about analysis, you say? Well, I wanted to end this by explaining what I consider one of the most beautiful things in mathematics, Grothendieck’s generalization of topology. There, instead of thinking of open sets as subsets of a space, you instead consider them as nice maps into a space. The intersection becomes a type of product (a “fiber product”) and one formalizes the notion of an open cover. You end up being able to put a topology on a category, and do all sorts of amazing and cool stuff. Unfortunately, I just couldn’t come up with a comprehensible explanation. But I hope the message is clear: in math, it’s not always about what you’re made of, but rather it’s about how you relate.