
In 1963, the PhD thesis of Francis William Lawvere (1937-2023) revolutionized how mathematicians look at algebraic structures, and kickstarted a new branch of mathematics. The main contribution of this thesis: Lawvere theories.
Not exactly a self-explanatory name for a concept, so let me explain. Prior to Lawvere’s thesis, the theories of algebraic structures (essentially, the operations and rules that characterize them) were presented as lists: a list of operations (also called the language or signature) and a list of equations that the operations had to obey. What Lawvere showed in his thesis was that this data could be encoded into a category: a Lawvere theory.
I dedicated a whole article to the concept of categories (which you should definitely go check out if you haven’t already), but as a recap, a category basically consists of a collection of nodes called objects, and arrows called morphisms that point from one object to another, relating the objects to each other (subject to a couple basic rules).
So how do we turn lists of operations and rules into a category? We can start by taking a closer look at what we mean by “operation”.
One way to visualize a mathematical operation is as a machine, one that takes in some inputs and produces one output. We should take note of what types of things the machine can take as inputs, and what types of outputs it produces. In general, this can get quite complicated, but for our purposes we can assume that, for any operation that we look at, all of its inputs and its output come from the same bucket of things (say, whole numbers). Take addition for example. Pick any two whole numbers, add them together, and your answer will inevitably be another whole number.
Addition, and probably the vast majority of the operations that you’re familiar with, takes two inputs (no matter how many numbers you may want to add together, you can only ever do it two at a time). We can think of the process of choosing these inputs as having two copies of this one bucket of things, and picking one element from each. This would also generalize to more complicated operations with many more inputs; we just start with more copies of this bucket, and pick one element from each. And since the outputs of our operations also belong to this same bucket, we can think of an operation as being a machine that connects, say, n copies of this bucket to a single copy of the bucket. We could even start building a sort of “factory line” by using the outputs of some machines as inputs for others!
So, if we define the objects of our category-to-be as collections of (finitely many) copies of this bucket (one copy is an object, two copies is another object, and so on), then our operations, and all of the larger factory lines that can be built, can naturally become arrows in this structure.
That’s the operations dealt with, but how does the list of rules come into play? Well, what the rules are effectively saying is which operations and factory lines are equal. So, we can incorporate the rules into our category-to-be just by having any two operations or factory lines that the rules say are equal be represented by the same arrow. It turns out that this makes our resulting structure a true-blue category: a Lawvere theory!
The fact that deep connections like this are possible using category theory is what drives my interest in the subject. But consider this: not all mathematical structures can be represented by just operations and rules, so basic Lawvere theories aren’t enough to be able to talk about them in a categorical way. For example, there is no Lawvere theory of categories. We need more tools at our disposal to be able to talk about these things; a more expressive categorical language, so to speak. The question is, what should that language be?
Photo by GDJ via Open Clipart