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## 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.

## 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.

## 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.

## Regular and effective monomorphisms and epimorphisms

Previously we observed that although monomorphisms tended to give expected generalizations of injective function in many categories, epimorphisms sometimes weren’t the expected generalization of surjective functions. We also discussed split epimorphisms, but where the definition of an epimorphism is too permissive to agree with the surjective morphisms in familiar concrete categories, the definition of a split epimorphism is too restrictive.

In this post we will discuss two other intermediate notions of epimorphism. (These all give dual notions of monomorphisms, but their epimorphic variants are more interesting as a possible solution to the above problem.) There are yet others, but these two appear to be the most relevant in the context of abelian categories.

## Groupoids

My current top candidate for a mathematical concept that should be and is not (as far as I can tell) consistently taught at the advanced undergraduate / beginning graduate level is the notion of a groupoid. Today’s post is a very brief introduction to groupoids together with some suggestions for further reading.

## Split epimorphisms and split monomorphisms

• What is the “easiest way” a morphism can be a monomorphism (resp. epimorphism)?
• What are the absolute monomorphisms (resp. epimorphisms) – that is, the ones which are preserved by every functor?
• A morphism which is both a monomorphism and an epimorphism is not necessarily an isomorphism. Can we replace either “monomorphism” or “epimorphism” by some other notion to repair this?
• If we wanted to generalize surjective functions, why didn’t we define an epimorphism to be a map which is surjective on generalized points?

The answer to all of these questions is the notion of a split monomorphism (resp. split epimorphism), which is the subject of today’s post.

## Noncommutative probability and group theory

There are, roughly speaking, two kinds of algebras that can be functorially constructed from a group $G$. The kind which is covariantly functorial is some variation on the group algebra $k[G]$, which is the free $k$-module on $G$ with multiplication inherited from the multiplication on $G$. The kind which is contravariantly functorial is some variation on the algebra $k^G$ of functions $G \to k$ with pointwise multiplication.

When $k = \mathbb{C}$ and when $G$ is respectively either a discrete group or a compact (Hausdorff) group, both of these algebras can naturally be endowed with the structure of a random algebra. In the case of $\mathbb{C}[G]$, the corresponding state is a noncommutative refinement of Plancherel measure on the irreducible representations of $G$, while in the case of $\mathbb{C}^G$, the corresponding state is by definition integration with respect to normalized Haar measure on $G$.

In general, some nontrivial analysis is necessary to show that the normalized Haar measure exists, but for compact groups equipped with a faithful finite-dimensional unitary representation $V$ it is possible to at least describe integration against Haar measure for a dense subalgebra of the algebra of class functions on $G$ using representation theory. This construction will in some sense explain why the category $\text{Rep}(G)$ of (finite-dimensional continuous unitary) representations of $G$ behaves like an inner product space (with $\text{Hom}(V, W)$ being analogous to the inner product); what it actually behaves like is a random algebra, namely the random algebra of class functions on $G$.

## Internal equivalence relations

For the last few weeks I’ve been working as a counselor at the PROMYS program. The program runs, among other things, a course in abstract algebra, which was a good opportunity for me to get annoyed at the way people normally introduce normal subgroups, which is via the following unmotivated

Definition: A subgroup $N$ of a group $G$ is normal if $gNg^{-1} \subset N$ for all $g \in G$.

It is then proven that normal subgroups are precisely the kernels $N = \phi^{-1}(e)$ of surjective group homomorphisms $\phi : G \to G/N$. In other words, they are precisely the subgroups you can quotient by and get another group. This strikes me as backwards. The motivation to construct quotient groups should come first.

Today I’d like to present an alternate conceptual route to this definition starting from equivalence relations and quotients. This route also leads to ideals in rings and, among other things, highlights the special role of the existence of inverses in the theory of groups and rings (in the latter I mean additive inverses). The categorical setting for this discussion is the notion of a kernel pair and of an internal equivalence relation in a category, but for the sake of accessibility we will not use this language explicitly.

## A less biased definition of a group

Here’s what seems like a silly question: what’s the universal group? That is, what’s the universal example of a set $G$ together with maps

$\displaystyle e : 1 \to G, m : G \times G \to G, i : G \to G$

satisfying the identities

1. $m(e, x) = m(x, e) = x$,
2. $m(x, i(x)) = m(i(x), x)$,
3. $m(x, m(y, z)) = m(m(x, y), z)$?

A moment’s reflection shows that there isn’t such a group; the existence of the groups $\mathbb{Z}^S$, where $S$ is an arbitrary set, shows that there exist groups of arbitrarily large cardinality, so no particular group can be universal.