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Suitably nice groupoids have a numerical invariant attached to them called groupoid cardinality. Groupoid cardinality is closely related to Euler characteristic and can be thought of as providing a notion of integration on groupoids.

There are various situations in mathematics where computing the size of a set is difficult but where that set has a natural groupoid structure and computing its groupoid cardinality turns out to be easier and give a nicer answer. In such situations the groupoid cardinality is also known as “mass,” e.g. in the Smith-Minkowski-Siegel mass formula for lattices. There are related situations in mathematics where one needs to describe a reasonable probability distribution on some class of objects and groupoid cardinality turns out to give the correct such distribution, e.g. the Cohen-Lenstra heuristics for class groups. We will not discuss these situations, but they should be strong evidence that groupoid cardinality is a natural invariant to consider.

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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.

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The center Z(G) of a group is an interesting construction: it associates to every group G an abelian group Z(G) in what is certainly a canonical way, but not a functorial way: that is, it doesn’t extend (at least in any obvious way) to a functor \text{Grp} \to \text{Ab} (unlike the abelianization G/[G, G]). We might wonder, then, exactly what kind of construction the center is.

Of course, it is actually not hard to come up with a rather general example of a canonical but not functorial construction: in any category C we may associate to an object c \in C its automorphism group \text{Aut}(c) or endomorphism monoid \text{End}(c)), and this is a canonical construction which again doesn’t extend in an obvious way to a functor C \to \text{Grp} or C \to \text{Mon}. (It merely reflects some special part of the bifunctor \text{Hom}(-, -).)

It turns out that the center can actually be thought of in terms of automorphisms (or endomorphisms), not of a group G, but of the identity functor G \to G, where G is regarded as a category with one object. This definition generalizes, and the resulting general definition has some interesting specializations. Moreover, an important general property is that the center is always abelian, and this has a very elegant proof.

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Let X be a finite CW complex with c_0 vertices, c_1 edges, and in general c_i different i-cells. The Euler characteristic

\displaystyle \chi(X) = \sum_{i \ge 0} (-1)^i c_i

is a fundamental invariant of X, and the observation that it is homotopy invariant is the appropriate generalization of Euler’s formula V - E + F = 2 = \chi(S^2) for a convex polyhedron. But where exactly does this expression come from? The modern story involves the homology groups H_i(X, \mathbb{Q}), but actually one can work on a more intuitive level characterized by the following slogan:

The Euler characteristic is a homotopy-invariant generalization of cardinality.

More precisely, the above expression for Euler characteristic can be deduced from three simple axioms:

  1. Cardinality: \chi(\text{pt}) = 1.
  2. Homotopy invariance: If X \sim Y, then \chi(X) = \chi(Y).
  3. Inclusion-exclusion: Suppose X is the union of two subcomplexes A, B whose intersection A \cap B is a subcomplex of both A and B. Then \chi(X) = \chi(A) + \chi(B) - \chi(A \cap B).

Of course, this isn’t enough to conclude that there actually exists an invariant with these properties. Nevertheless, it’s enough to motivate the search for a proof that such an invariant exists.

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SU(2) and the quaternions

The simplest compact Lie group is the circle S^1 \cong \text{SO}(2). Part of the reason it is so simple to understand is that Euler’s formula gives an extremely nice parameterization e^{ix} = \cos x + i \sin x of its elements, showing that it can be understood either in terms of the group of elements of norm 1 in \mathbb{C} (that is, the unitary group \text{U}(1)) or the imaginary subspace of \mathbb{C}.

The compact Lie group we are currently interested in is the 3-sphere S^3 \cong \text{SU}(2). It turns out that there is a picture completely analogous to the picture above, but with \mathbb{C} replaced by the quaternions \mathbb{H}: that is, \text{SU}(2) is isomorphic to the group of elements of norm 1 in \mathbb{H} (that is, the symplectic group \text{Sp}(1)), and there is an exponential map from the imaginary subspace of \mathbb{H} to this group. Composing with the double cover \text{SU}(2) \to \text{SO}(3) lets us handle elements of \text{SO}(3) almost as easily as we handle elements of \text{SO}(2).

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SO(3) and SU(2)

In order to study the hydrogen atom, we’ll need to know something about the representation theory of the special orthogonal group \text{SO}(3). This post consists of a few preliminaries along the way to doing this. I’ll be somewhat vague about a few things that 1) I don’t have much experience with, and 2) that would detract from the main narrative anyway.

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The following two lemmas might be encountered in a basic course in complex analysis (the first in a basic course in group theory, even).

Lemma 1: Fix a field F. The group of fractional linear transformations PGL_2(F) acts triple transitively on \mathbb{P}^1(F) and the stabilizer of any triplet of distinct points is trivial.

Lemma 2: The group of fractional linear transformations on \mathbb{P}^1(\mathbb{C}) preserving the upper half plane \mathbb{H} = \{ z \in \mathbb{C} | \text{Im}(z) > 0 \} is PSL_2(\mathbb{R}).

I used to only know extremely boring computational proofs of both of these statements. However, I now know better! Today I’d like to give shorter and conceptual proofs of both of these, and then briefly discuss how they come about in the study of elliptic curves (a subject I’d like to talk about in more detail once this semester is over).

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