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## The cohomology of the n-torus

The goal of this post is to compute the cohomology of the $n$-torus $X = (S^1)^n \cong \mathbb{R}^n/\mathbb{Z}^n$ in as many ways as I can think of. Below, if no coefficient ring is specified then the coefficient ring is $\mathbb{Z}$ by default. At the end we will interpret this computation in terms of cohomology operations.

## Cartesian closed categories and the Lawvere fixed point theorem

Previously we saw that Cantor’s theorem, the halting problem, and Russell’s paradox all employ the same diagonalization argument, which takes the following form. Let $X$ be a set and let

$\displaystyle f : X \times X \to 2$

be a function. Then we can write down a function $g : X \to 2$ such that $g(x) \neq f(x, x)$. If we curry $f$ to obtain a function

$\displaystyle \text{curry}(f) : X \to 2^X$

it now follows that there cannot exist $x \in X$ such that $\text{curry}(f)(x) = g$, since $\text{curry}(f)(x)(x) = f(x, x) \neq g(x)$.

Currying is a fundamental notion. In mathematics, it is constantly implicitly used to talk about function spaces. In computer science, it is how some programming languages like Haskell describe functions which take multiple arguments: such a function is modeled as taking one argument and returning a function which takes further arguments. In type theory, it reproduces function types. In logic, it reproduces material implication.

Today we will discuss the appropriate categorical setting for understanding currying, namely that of cartesian closed categories. As an application of the formalism, we will prove the Lawvere fixed point theorem, which generalizes the argument behind Cantor’s theorem to cartesian closed categories.

## The homotopy groups are only groups

Often in mathematics we define constructions outputting objects which a priori have a certain amount of structure but which end up having more structure than is immediately obvious. For example:

• Given a Lie group $G$, its tangent space $T_e(G)$ at the identity is a priori a vector space, but it ends up having the structure of a Lie algebra.
• Given a space $X$, its cohomology $H^{\bullet}(X, \mathbb{Z})$ is a priori a graded abelian group, but it ends up having the structure of a graded ring.
• Given a space $X$, its cohomology $H^{\bullet}(X, \mathbb{F}_p)$ over $\mathbb{F}_p$ is a priori a graded abelian group (or a graded ring, once you make the above discovery), but it ends up having the structure of a module over the mod-$p$ Steenrod algebra.

The following question suggests itself: given a construction which we believe to output objects having a certain amount of structure, can we show that in some sense there is no extra structure to be found? For example, can we rule out the possibility that the tangent space to the identity of a Lie group has some mysterious natural trilinear operation that cannot be built out of the Lie bracket?

In this post we will answer this question for the homotopy groups $\pi_n(X)$ of a space: that is, we will show that, in a suitable sense, each individual homotopy group $\pi_n(X)$ is “only a group” and does not carry any additional structure. (This is not true about the collection of homotopy groups considered together: there are additional operations here like the Whitehead product.)

## The p-group fixed point theorem

The goal of this post is to collect a list of applications of the following theorem, which is perhaps the simplest example of a fixed point theorem.

Theorem: Let $G$ be a finite $p$-group acting on a finite set $X$. Let $X^G$ denote the subset of $X$ consisting of those elements fixed by $G$. Then $|X^G| \equiv |X| \bmod p$; in particular, if $p \nmid |X|$ then $G$ has a fixed point.

Although this theorem is an elementary exercise, it has a surprising number of fundamental corollaries.

## Cantor’s theorem, the prisoner’s dilemma, and the halting problem

Cantor’s theorem is somewhat infamous as a mathematical result that many non-mathematicians have a hard time believing. Trying to disprove Cantor’s theorem is a popular hobby among students and cranks; even Eliezer Yudkowsky1993 fell into this trap once. I think part of the reason is that the standard proof is not very transparent, and consequently is hard to absorb on a gut level.

The goal of this post is to present a rephrasing of the statement and proof of Cantor’s theorem so that it is no longer about sets, but about a particular kind of game related to the prisoner’s dilemma. Rather than showing that there are no surjections $X \to 2^X$, we will show that a particular kind of player in this game can’t exist. This rephrasing may make the proof more transparent and easier to absorb, although it will take some background material about the prisoner’s dilemma to motivate. As a bonus, we will almost by accident run into a proof of the undecidability of the halting problem.

## Operations, pro-objects, and Grothendieck’s Galois theory

Previously we looked at several examples of $n$-ary operations on concrete categories $(C, U)$. In every example except two, $U$ was a representable functor and $C$ had finite coproducts, which made determining the $n$-ary operations straightforward using the Yoneda lemma. The two examples where $U$ was not representable were commutative Banach algebras and commutative C*-algebras, and it is possible to construct many others. Without representability we can’t apply the Yoneda lemma, so it’s unclear how to determine the operations in these cases.

However, for both commutative Banach algebras and commutative C*-algebras, and in many other cases, there is a sense in which a sequence of objects approximates what the representing object of $U$ “ought” to be, except that it does not quite exist in the category $C$ itself. These objects will turn out to define a pro-object in $C$, and when $U$ is pro-representable in the sense that it’s described by a pro-object, we’ll attempt to describe $n$-ary operations $U^n \to U$ in terms of the pro-representing object.

The machinery developed here is relevant to understanding Grothendieck’s version of Galois theory, which among other things leads to the notion of étale fundamental group; we will briefly discuss this.

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.

## 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.