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Previously we described $n$-ary operations on (the underlying sets of the objects of) a concrete category $(C, U)$, which we defined as the natural transformations $U^n \to U$.

Puzzle: What are the $n$-ary operations on finite groups?

Note that $U$ is not representable here. The next post will answer this question, but for those who don’t already know the answer it should make a nice puzzle.

## Operations and Lawvere theories

Groups are in particular sets equipped with two operations: a binary operation (the group operation) $(x_1, x_2) \mapsto x_1 x_2$ and a unary operation (inverse) $x_1 \mapsto x_1^{-1}$. Using these two operations, we can build up many other operations, such as the ternary operation $(x_1, x_2, x_3) \mapsto x_1^2 x_2^{-1} x_3 x_1$, and the axioms governing groups become rules for deciding when two expressions describe the same operation (see, for example, this previous post).

When we think of groups as objects of the category $\text{Grp}$, where do these operations go? They’re certainly not morphisms in the corresponding categories: instead, the morphisms are supposed to preserve these operations. But can we recover the operations themselves?

It turns out that the answer is yes. The rest of this post will describe a general categorical definition of $n$-ary operation and meander through some interesting examples. After discussing the general notion of a Lawvere theory, we will then prove a reconstruction theorem and then make a few additional comments.

## The type system of mathematics

Occasionally I see mathematical questions that seem “grammatically incorrect” in some sense.

Example. “Is $[0, 1]$ open or closed?”
Example. “Is $\{ 1, 2, 3 \}$ a group?”
Example. “What’s the Fourier series of $\sin x + \sin \pi x$?”

Here are some sillier examples.

Example. “Is a rectangle prime?”
Example. “Is $17 \in 3$?”
Example. “What’s the Fourier series of the empty set?”

What all of these examples have in common is that they are type errors: they are attempts to apply some mathematical process to a kind of mathematical object it was never intended to take as input. If you tried to write a program in some highly mathematical programming language to answer these questions, it (hopefully!) wouldn’t compile.

Mathematical objects are usually not explicitly thought of as having types in the same way that objects in a programming language with a type system has types. Ordinary mathematics is supposed to be formalizable within Zermelo-Fraenkel (ZF) set theory, possibly with the axiom of choice, and in ZF every mathematical object is constructed as a set. In that sense they all have the same type. (In particular, the question “is $17 \in 3$?” is perfectly meaningful in ZF! This is one reason not to like ZF as a foundation of mathematics.) However, I think that in practice mathematical objects are implicitly thought of as having types, and that this is a mental habit mathematicians pick up but don’t often talk about.

Instead of thinking in terms of set theory, thinking of mathematical objects as having types allows us to import various useful concepts into mathematics, such as the notions of type safety, typecasting, subtyping, and overloading, that help us make more precise what we mean by a mathematical sentence being “grammatically incorrect.” The rest of this post will be a leisurely discussion of these and other type-based concepts as applied to mathematics in general. There are various categorical ideas here, but for the sake of accessibility we will restrict them to parenthetical remarks.

## Constructions vs. specifications

What does it mean to define a mathematical object?

Roughly speaking, there are two strategies you could use. One is to construct that object from other objects. For example, $\mathbb{R}$ can be constructed from $\mathbb{Q}$ using Cauchy sequences or Dedekind cuts.

Another is to give a list of properties that uniquely specify the object. For example, $\mathbb{R}$ can be defined as the unique Dedekind-complete totally ordered field.

Let’s call these strategies construction and specification. Construction has the advantage of concreteness, and it also proves that the object exists. However, it has the disadvantage that constructions are not unique in general and you might not be working with the most useful construction at a given time; to prove things about an interesting mathematical object you might have to switch between various constructions (while also proving that they all construct the same object).

Relying excessively on construction also has a curious disadvantage when talking to students about the object in question: namely, they might ask you how to show that object $X$ satisfies property $P$, and you might respond “well, it’s obvious if you use construction $A$,” and they might respond “but I was taught that $X$ is [construction $B$]!” The possibility of multiple different constructions of the same mathematical object makes construction somewhat unsatisfying in terms of describing what you even mean when you say “object $X$.”

This is the appeal of specification. Although specification is more abstract and harder to grasp at first, it has the advantage you don’t have to use any particular construction of an object to prove things about it: instead, you should in principle be able to prove everything from the properties alone, since those uniquely specify the object. This often leads to cleaner proofs. Specifications also usually give a better motivation for looking at an object: it’s easier to make sense of the claim that we want a mathematical object with various properties than to make sense of the claim that we want a mathematical object which is constructed in some arbitrary-looking (to the uninitiated) fashion.

Below we’ll go through a few examples of mathematical objects and some constructions and specifications of them.

## Connected objects and a reconstruction theorem

A common theme in mathematics is to replace the study of an object with the study of some category that can be built from that object. For example, we can

• replace the study of a group $G$ with the study of its category $G\text{-Rep}$ of linear representations,
• replace the study of a ring $R$ with the study of its category $R\text{-Mod}$ of $R$-modules,
• replace the study of a topological space $X$ with the study of its category $\text{Sh}(X)$ of sheaves,

and so forth. A general question to ask about this setup is whether or to what extent we can recover the original object from the category. For example, if $G$ is a finite group, then as a category, the only data that can be recovered from $G\text{-Rep}$ is the number of conjugacy classes of $G$, which is not much information about $G$. We get considerably more data if we also have the monoidal structure on $G\text{-Rep}$, which gives us the character table of $G$ (but contains a little more data than that, e.g. in the associators), but this is still not a complete invariant of $G$. It turns out that to recover $G$ we need the symmetric monoidal structure on $G\text{-Rep}$; this is a simple form of Tannaka reconstruction.

Today we will prove an even simpler reconstruction theorem.

Theorem: A group $G$ can be recovered from its category $G\text{-Set}$ of $G$-sets.

## The Summer Program on Applied Rationality and Cognition

This summer I will be teaching at a newish high school summer math program, the Summer Program on Applied Rationality and Cognition (SPARC). We’ll be covering a wide range of topics, including probability, Bayesian statistics, and cognitive science, with the general theme of learning how to make rational decisions (both practically and theoretically). Many interesting people are involved, and I’m excited to see how the program will go.

I think SPARC will be an extremely valuable experience for talented high school students. If you are (resp. know of) such a student, I strongly encourage you to apply (resp. forward this information to them so that they can apply)! Questions about the program not addressed in the FAQ should be directed to contact@sparc2013.org.

## Update

I’ve uploaded notes for the classes I’m taking this semester again. This semester I’m taking the following:

• C*-algebras (Rieffel): An introduction to C*-algebras from the noncommutative geometry point of view. Should be quite interesting.
• Discrete Mathematics for the Life Sciences (Pachter): An introduction to computational genomics. I’m hoping to learn something about what kind of mathematics get used in biology.
• Algebraic Geometry (Nadler): Algebraic geometry from the point of view of categories of (quasi)coherent sheaves, their derived categories, etc. Should also be quite interesting.