• replace the study of a group with the study of its category of linear representations,
• replace the study of a ring with the study of its category of -modules,
• replace the study of a topological space with the study of its category 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 is a finite group, then as a category, the only data that can be recovered from is the number of conjugacy classes of , which is not much information about . We get considerably more data if we also have the monoidal structure on , which gives us the character table of (but contains a little more data than that, e.g. in the associators), but this is still not a complete invariant of . It turns out that to recover we need the symmetric monoidal structure on ; this is a simple form of Tannaka reconstruction.

Today we will prove an even simpler reconstruction theorem.

Theorem: A group can be recovered from its category of -sets.

Connected objects

The idea of the proof is that we want to find a categorical property that allows us to isolate the subcategory of transitive -sets. The -set itself can be uniquely identified among transitive -sets as the “largest” one (more precisely, it is the unique weak initial object among transitive -sets), and then can be recovered as the opposite group of the group of automorphisms of (as a -set).

The categorical property we want is the following.

Definition: An object in a category is connected if the representable functor preserves coproducts.

The idea behind the definition is that if one thinks of coproducts as a disjoint union, then a morphism from a connected object into a disjoint union of objects must land entirely in one of the objects, or else it will be “disconnected” by the fact that it’s spread out over a disjoint union.

Example. In , the connected objects are precisely the one-element sets. Note that the empty set is not connected.

Example. In (which we’ll take to be the category of simple graphs), the connected objects are precisely the connected graphs in the usual sense. Note that the empty graph is not connected.

Example. In , the connected objects are precisely the connected topological spaces in the usual sense. Note that the empty space is not connected.

Example. In , any affine scheme such that has nontrivial idempotents (in other words, such that is not a connected ring) is not connected in the categorical sense. To see this, let contain a nontrivial idempotent and consider the map induced by the ring homomorphism

.

This map factors through neither of the projections , from which it follows that is not connected. (One interpretation of this argument is that is the free commutative ring on an idempotent.)

The converse statement – that the spectrum of a connected ring is connected in the categorical sense – is false. For example, let be a homomorphism induced by a non-principal ultrafilter on an infinite set . Then the corresponding morphism does not factor through any of the inclusions, so is not connected in the categorical sense. More generally, any ring which can be obtained as a nontrivial ultraproduct does not have connected spectrum in the categorical sense. However, it is true that if is a connected ring, then preserves finite coproducts.

Note that the spectrum of the zero ring is not connected. In general, the initial object of a category is never connected; it is too simple to be simple.

Intuitively, a nontrivial coproduct should not be connected. The following result shows that this is true under reasonable hypotheses. These hypotheses don’t hold for but do hold for , which is enough; I don’t know how much they can be relaxed.

Proposition: Let be two objects, neither of which is the initial object, of a concrete category with finite coproducts such that the forgetful functor preserves finite coproducts. Then is not connected.

Proof. We prove the contrapositive, namely that under the hypotheses above, if is connected then one of is the initial object. Recall that “preserves coproducts” means the following, for binary coproducts. If are objects, the natural inclusions induce natural maps which in turn induces a natural map

and to say that preserves binary coproducts means that this map is always a bijection for all . In particular, every morphism factors through one of the inclusions .

Now let . Suppose that is connected; then WLOG the identity map factors through . It follows that the inclusion map is a split epimorphism. However, by assumption, the map on underlying sets is an injection in , and since faithful functors reflect monomorphisms, it follows that the inclusion map is a monomorphism, hence an isomorphism. But looking at the corresponding natural isomorphism of representable functors , this is possible if and only if is the initial object.

(Note that a sufficient condition for to preserve finite coproducts is that it has a right adjoint. This is true of the forgetful functors from , and to .)

Reconstruction from

Proposition: The connected objects of are precisely the transitive -sets (the empty -set is not transitive).

Proof. Every -set can be expressed uniquely as a coproduct of transitive -sets (the orbits of the group action). It follows that a connected object of is necessarily transitive. Conversely, if is a transitive -set, then the image of any homomorphism from into another -set is necessarily also a transitive -set, hence contained in an orbit of .

Proposition: The -set is the unique (up to isomorphism) transitive -set which admits a morphism to all other transitive -sets.

Proof. Every transitive -set has the form for some subgroup of , so in particular admits a quotient map . On the other hand, if is not the trivial subgroup, then there exist no morphisms , since no map of sets can respect the -action (nontrivial elements of preserve the identity coset of but can’t preserve its image in ).

Theorem: A group can be recovered from its category of -sets.

Proof. We know that from the category of -sets we can recover the subcategory of transitive -sets, and we know that from the transitive -sets we can recover the -set itself as the unique weak initial object. The automorphism group of , as a -set, is , from which we can recover by taking the opposite group.

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.

• 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.

• really should be done with diagrams (and it’s not easy to finish a blog post with diagrams in a day) or
• might take more time than I allot for blogging in a day to work through the relevant concepts.

Sticking to one post a day at this point is likely to drive down quality, so I think I am going to stop doing it. It was a good goal for awhile in that it got me to write some posts that I’d wanted to write for a long time now, but unfortunately it is now doing the opposite of that.

Part of Schur-Weyl duality asserts that the answer is yes. The double commutant theorem plays an important role in the proof and also highlights an important corollary, namely that admits a canonical decomposition

where runs over partitions, are some irreducible representations of , and are the Specht modules, which describe all irreducible representations of . This gives a fundamental relationship between the representation theories of the general linear and symmetric groups; in particular, the assignment can be upgraded to a functor called a Schur functor, generalizing the construction of the exterior and symmetric products.

The proof below is more or less from Etingof’s notes on representation theory (Section 4.18). We will prove four versions of Schur-Weyl duality involving , and (in the special case that is a complex inner product space) .

Throughout this post, denotes a finite-dimensional vector space over a field of characteristic .

Schur-Weyl duality for

Before proving Schur-Weyl duality for the Lie group , we first prove it for the Lie algebra . First, recall that if are representations of a Lie algebra , then their tensor product is also a representation of where acts by the Leibniz rule:

.

This ensures that the exponential acts the way an element of a group acts on a tensor product:

.

Second, we will need some preparatory lemmas.

Lemma 1: Suppose that for all , contains the element

where are vectors in some vector space containing . Then contains .

Proof. We proceed by induction on the degree . The result is clear for . In general, setting we conclude that contains . It follows that contains

for all . But over an infinite field, any value of a polynomial of degree is a linear combination of values that the polynomial attains at any distinct points by Lagrange interpolation. It follows that contains for all , which is a polynomial of degree , and the conclusion follows by induction.

Lemma 2: The symmetric power is spanned by elements of the form .

Proof. Let be the subspace of spanned by elements of the form and let be a basis of . Then contains the elements

for all choices of scalars . Applying Lemma 1 times, it follows that contains for all choices of such that , and these form a basis of . (We need the hypothesis that has characteristic zero or else some of the coefficients above may vanish.)

Lemma 3: Let be a finite-dimensional algebra over a field of characteristic . Then the invariant subalgebra is generated as an algebra by the elements

.

Proof. By the fundamental theorem of symmetric functions, the elementary symmetric function is some polynomial in the power symmetric functions as ranges from to . We conclude that

is a polynomial in the elements

.

Over a field of characteristic we can freely identify with , and then the conclusion follows by Lemma 2.

We are now ready to prove the theorem.

Theorem (Schur-Weyl duality): The natural map is surjective.

Proof. Elements act on by the Leibniz rule as described above:

.

Let be the subalgebra of spanned by operators like the above (equivalently, the image of the universal enveloping algebra in . We want to show that the commutant of is the image of in . By Maschke’s theorem, is a finite-dimensional semisimple -module, so by the double commutant theorem it suffices to show instead that the commutant of the image of is .

The commutant of the image of in is precisely the invariant subalgebra

with respect to the natural action of on . By Lemma 3, is generated by elements of the form

and the conclusion follows.

Corollary (existence of Schur functors): Let be a partition describing an irreducible representation of the symmetric group . Then there is a functor such that is either an irreducible representation of or zero.

To make sense of this result it is not necessary to understand in detail the classification of the irreducible representations of , but it is helpful to know that they can all be realized over , and consequently the description of the Schur functors above does not depend on the base field .

Example. If is the trivial representation of , then is the space of symmetric tensors . In characteristic this functor is naturally isomorphic to the symmetric power functor (which is constructed as a quotient rather than a subspace of ), but in positive characteristic the two are not isomorphic as -representations.

Example. If is the sign representation of , then is the space of antisymmetric tensors . In characteristic this functor is naturally isomorphic to the exterior power functor (which is constructed as a quotient rather than a subspace of ), but in positive characteristic the definitions of symmetric and antisymmetric tensor coincide, although there is still a usable salvage of the exterior power functor.

Schur-Weyl duality for

Theorem: The subalgebra of spanned by elements of is precisely the subalgebra of spanned by elements of .

Proof. Since the subalgebra spanned by is contained in the commutant of the image of , by Schur-Weyl duality for it is contained in the subalgebra spanned by . To show the reverse inclusion, observe that if then is in the subalgebra spanned by for all but finitely many . But by Lagrange interpolation it must in fact lie in this subalgebra for all , in particular for . The result then follows from Lemma 2.

Corollary (Schur-Weyl duality): The natural map is surjective.

Corollary: is either an irreducible representation of or zero.

Schur-Weyl duality for

We now take and equip with an inner product. This allows us to replace with the unitary Lie algebra

of skew-adjoint operators because we have a natural isomorphism . More explicitly, are precisely the self-adjoint operators, and any element is uniquely a linear combination of a skew-adjoint and a self-adjoint operator, namely

.

In other words, is a real form of . It is not the only real form; if , then is of course also a real form.

In any case, it follows that the span of the image of in the space of endomorphisms of any representation functorially constructed from is the same as the span of the image of .

Corollary (Schur-Weyl duality): The natural map is surjective.

Corollary: is either an irreducible representation of or zero.

Schur-Weyl duality for

As for the general linear group, we can now lift the previous result to the unitary group .

Theorem: The subalgebra of spanned by elements of is precisely the subalgebra of spanned by elements of .

Proof. Given , we observed previously that the exponential of the action of on is the action of the exponential of on . Since the exponential map is surjective, this establishes one inclusion. To establish the other, given we can differentiate the action of to obtain the action of .

Corollary (Schur-Weyl duality): The natural map is surjective.

Corollary: is either an irreducible representation of or zero.

,

as the composite of some kind of epimorphism and some kind of monomorphism . Here we should think of as something like the image of . This is most familiar, for example, in the case of , and other algebraic categories, where is the set-theoretic image of in the usual sense.

Today we will discuss some general properties of factorizations of a morphism into an epimorphism followed by a monomorphism, or epi-mono factorizations. The failure of such factorizations to be unique turns out to be closely related to the failure of epimorphisms or monomorphisms to be regular.

The category of factorizations

Define the category of epi-mono factorizations of a morphism to be the category whose objects are epi-mono factorizations

of (so is an epimorphism, is a monomorphism, and ) and whose morphisms are morphisms making the diagram

commute; that is, such that and . These two properties are already enough to conclude the following:

1. is an epimorphism (since is an epimorphism).
2. is a monomorphism (since is a monomorphism).
3. If exists, it is unique (since is an epimorphism, or alternately since is a monomorphism).

Thus the category of epi-mono factorizations of a morphism is a preorder. Moreover, the morphisms in the category are both monomorphisms and epimorphisms. Call such a morphism a fake isomorphism if it is not an isomorphism (this terminology is nonstandard).

If we are working in a category with no fake isomorphisms, such as or , then any two epi-mono factorizations which are related by a morphism are isomorphic via a unique isomorphism. (This doesn’t rule out the possibility that there are two epi-mono factorizations which are not related by any morphisms at all.) However, because there are categories with fake isomorphisms, we do not expect uniqueness of epi-mono factorizations in general.

Example. In , let be an integral domain and let be the inclusion of into its field of fractions. If is not a field, then is a fake isomorphism; moreover, the category of epi-mono factorizations of is equivalent to the poset of subrings of containing , or equivalently the poset of localizations of .

Example. In , any continuous bijection which does not have a continuous inverse is a fake isomorphism. Without loss of generality, we may take and to have the same underlying set; then we are just talking about a pair of topologies on one of which is strictly finer than the other. The category of epi-mono factorizations of is then equivalent to the poset of topologies intermediate between these two topologies.

More generally, if is a fake isomorphism, then it admits two nonisomorphic factorizations

.

So the problem of non-uniqueness of epi-mono factorizations is closely related to the problem of existence of fake isomorphisms. Furthermore, previously we showed that a morphism which is either both a monomorphism and a regular epimorphism or which is both a regular monomorphism and an epimorphism is necessarily an isomorphism. It follows conversely that the existence of fake isomorphisms indicates the existence of epimorphisms or monomorphisms which are not regular.

Besides uniqueness, in full generality it is also necessary to worry about existence. For example, consider the free category on an idempotent. This is a category with a single object and a single non-identity morphism satisfying . Then is neither a monomorphism nor an epimorphism, since the above identity shows that it is neither left nor right cancellable, and since the only possible factorizations of are as

it follows that does not admit an epi-mono factorization.

Connectivity

Above we observed that one issue with the category of epi-mono factorizations is that it may fail to be connected: that is, there may be two epi-mono factorizations that are not related by any chain of morphisms, hence even if there were no fake isomorphisms we would still not be able to conclude that epi-mono factorizations are unique.

However, mild categorical hypotheses guarantee that this is not an issue.

Theorem: Suppose that a category has either pushouts or pullbacks. Moreover, suppose that has no fake isomorphisms (e.g. because all monomorphisms are regular or because all epimorphisms are regular). Then epi-mono factorizations in are unique (up to unique isomorphism).

Proof. The second hypothesis and the conclusion are categorically self-dual but the first hypothesis is not, so it suffices to prove the statement under the assumption that has pushouts. If and are two epi-mono factorizations of a morphism , consider the pushout together with the inclusions and induced map :

We claim that is an epimorphism. To see this, suppose are two other morphisms such that

.

Since are epimorphisms, it follows that and . Hence and describe the same commutative square, from which it follows by the universal property of the pushout that they factor through the same morphism , namely .

It follows that are both epimorphisms. On the other hand, since and are monomorphisms, it follows that are both monomorphisms. Since has no fake isomorphisms, it follows that are both isomorphisms, hence is a monomorphism and determines an epi-mono factorization which is isomorphic to both and . The conclusion follows.

Corollary: Suppose is a category with either pushouts or pullbacks. Then epi-mono factorizations in are unique if and only if has no fake isomorphisms.

Note that the corollary does not say anything about existence.

.

The commutant of is equal to the commutant of the subring of generated by the , so we may assume without loss of generality that is already such a subring. In that case, is just the ring of endomorphisms of as a left -module. The use of the term commutant instead can be thought of as emphasizing the role of and de-emphasizing the role of .

The assignment is a contravariant Galois connection on the lattice of subsets of , so the double commutant may be thought of as a closure operator. Today we will prove a basic but important theorem about this operator.

Warmup: multiplicities

If is a finite group and a finite-dimensional complex representation of , then breaks up into a direct sum

of irreducible representations with some multiplicities . However, this direct sum decomposition is not canonical if the multiplicities are greater than . In the worst case, may act trivially on , and then is a direct sum of copies of the trivial representation. Actually choosing such a direct sum decomposition is equivalent to choosing a basis of .

However, there is an alternate and completely canonical way of describing a representation in terms of its irreducible subrepresentations without choosing a direct sum decomposition as above. As a first hint, note that

.

This suggests that it might be useful to replace with the vector space . And, in fact, this turns out to be a great idea: there is a canonical evaluation map

whose image is precisely the -isotypic component of , and this gives an alternate canonical decomposition of as

which does not require making any choices. One can think of as the multiplicity space associated to , the correct canonical replacement for the multiplicity .

The idea of the double commutant theorem is to think about what kind of structure multiplicity spaces have. So far we have been using them only as vector spaces, but in fact they are -modules. Note that is precisely the commutant of the image of in .

Basic properties of commutants

Now that our warmup is done, we list some basic properties of the commutant operation

.

1. is a subring of .
2. implies .
3. if and only if .
4. (by 3).
5. (by 2 and 4).
6. (by 4 and 5).

The second and third properties assert that the commutant establishes a special type of Galois connection. In the language of category theory, the second and third properties assert that the commutant is a contravariant functor from the poset of subsets of to itself which happens to be its own adjoint. The remaining properties verify something slightly stronger than the statement that the double commutant is a closure operator: they also verify that the subsets of which are their own double commutant are precisely the commutants of other subsets of .

The double commutant theorem

Theorem (double commutant): Let be an abelian group and let be a subring of such that

1. is a semisimple ring, and
2. is a finite direct sum of simple -modules.

Then is its own double commutant. Moreover, is also semisimple, and as a -module, decomposes as a direct sum

where is a complete list of the simple -modules, is a complete list of the simple -modules, and . In particular, there is a canonical bijection between simple -modules and simple -modules.

Proof. Choose a finite direct sum decomposition

where the are the simple -modules. Since acts faithfully on , it follows (for example by Artin-Wedderburn) that the multiplicities are all positive. By Schur’s lemma,

where are division rings; in particular, is semisimple. Now, acts on the multiplicity spaces , and by inspection of the two decompositions above these are precisely the simple -modules. More precisely, is the unique simple
-module on which the factor acts nontrivially, and it is in particular an -dimensional -vector space (since acts on on the left, it acts on on the right). Hence, as in the finite group case above, the natural map

is an isomorphism. Writing , we may now think of the as the multiplicity spaces of the decomposition of as a -module, we conclude that is also the finite direct sum of simple -modules (with multiplicities given by the dimensions of the as -vector spaces), and it follows from here that by Artin-Wedderburn.

If you don’t like division rings, feel free to assume that is a finite-dimensional vector space over an algebraically closed field , which case everything above is a -vector space.

Example. Let be a finite group and a subgroup, and consider the representation of . The double commutant theorem tells us that decomposes into a direct sum as

where the are irreducible representations of and the are a complete list of the simple -modules. Understanding thus gives us a information about the decomposition of as a -module.

is one definition of the Hecke algebra . It may be described explicitly as spanned by double cosets , which have a well-defined product by a left coset on the left as a left coset. This construction is morally responsible for many of the Hecke algebras appearing in mathematics by making particular choices of and (usually and are not finite groups and so one passes from to a suitable space of functions on , but the idea is the same).

Theorem: Let be a prime power, let be a positive integer, and consider the distribution of irreducible factors of degree in a random monic polynomial of degree over . Then, as , this distribution is asymptotically the distribution of cycles of length in a random permutation of elements.

One can even name what this random permutation ought to be: namely, it is the Frobenius map acting on the roots of a random polynomial , whose cycles of length are precisely the factors of degree of .

Combined with our previous result, we conclude that as (with tending to infinity sufficiently quickly relative to ), the distribution of irreducible factors of degree is asymptotically independent Poisson with parameters .

Proof modulo some analytic details

By the cyclotomic identity, we have

where the sum runs over all monic polynomials over , the product runs over over all monic irreducible polynomials over , and

is the number of monic irreducible polynomials of degree over . This is Euler product for the zeta function for . To get the results we want we need an analogue for polynomials over finite fields of the exponential formula which keeps track of how many factors of a given degree a polynomial has. Letting denote the number of irreducible factors of degree of , this is given by

.

To get a generating function describing the joint moment generating function of the variables we will also need to divide the terms corresponding to polynomials of degree by the number of such polynomials, which is . This is equivalent to replacing with , which gives

.

We are now in a position to take the limit . We will do so somewhat cavalierly: since as and is fixed, we expect that

as , hence the above generating in some sense approaches

as , at least in the sense that their lower-order terms should match up reasonably well (where “lower-order” depends on ). But this is precisely

by the exponential formula.

Some comments on Granville’s analogy

The tripartite analogy between cycle decomposition of permutations, prime factorization of polynomials over a finite field, and prime factorization of integers can be made more specific as follows. There is a lot to say on this subject, but we will content ourselves with two sets of analogies.

First, the following three statements are analogous.

1. A random permutation of elements is a -cycle with probability .
2. A random monic polynomial of degree over is irreducible with probability asymptotically (as ).
3. An integer of size approximately is prime with probability approximately (the prime number theorem).

This suggests that the correct analogue of the degree of a positive integer is , an idea which is also suggested by the fact that the norm of a polynomial of degree over is , since this is the size of , whereas the norm of a positive integer is .

Second, the following three statements are analogous.

1. The distribution of cycles of length of a random permutation of elements is asymptotically (as ) independent Poisson with parameters .
2. The distribution of irreducible factors of degree of a random monic polynomial over of degree is asymptotically (as with growing sufficiently fast relative to ) independent Poisson with parameters .
3. The distribution of prime factors in the intervals of a random integer in the interval is asymptotically (as ) independent Poisson with parameters asymptotic to the sequence .

The third statement in this analogy is false as stated; for example, the distribution of prime factors in the interval is asymptotically geometric, not Poisson. It may be true if one takes in addition to in a suitable way while throwing out small primes in a way dependent on and while changing “independent Poisson” to “independent asymptotically Poisson,” where this second use of “asymptotically” depends on .

I offer as weak evidence the following heuristic calculation. First, the probability that a large positive integer is divisible by some is . If is a random variable which takes on non-negative integer values, then a straightforward telescoping argument shows that

.

If is the exponent of a prime in the prime factorization of , then heuristically , so

.

There is a well-known asymptotic for the sum of the reciprocals of the primes which implies that

and consequently the expected number of prime factors (counted with multiplicity) in the interval of a large positive integer is heuristically asymptotically .

(Edit, 11/11/12:) The heuristic calculation can be taken further as follows. Letting denote the random variable describing the exponent of in a random prime factorization, we have heuristically and therefore heuristically the moment generating function

.

For prime factorizations of large positive integers we expect the variables to heuristically be asymptotically independent, so heuristically their sum has moment generating function

.

By the PNT, there are approximately primes in this interval. Let us blithely pretend that they all have absolute value approximately , because this is somewhere between and and gives the expected value we computed above. Then we are reduced to a calculation very similar to the calculation over finite fields. Namely, we get a moment generating function

and as this is asymptotic to

which is the moment generating function of a Poisson random variable with parameter as expected.

Theorem: As , the number of cycles of length of a random permutation of elements are asymptotically independent Poisson with parameters .

This is a fairly strong statement which essentially settles the asymptotic description of short cycles in random permutations.

Proof

We will prove pointwise convergence of moment generating functions. First, the Poisson random variable with parameter is the random variable which takes on non-negative integer values with probabilities

.

therefore has moment generating function

which is the exponential generating function of the Touchard polynomials

where are the Stirling numbers of the second kind. These specialize to the Bell numbers when as expected.

Because we are discussing random variables, we should compute a joint moment generating function. The joint moment generating function of independent Poisson random variables with parameters is

.

Back to permutations. By the exponential formula, letting denote the number of cycles of length in a permutation , we have

which simplifies to

.

It is a general and straightforward observation that if is a power series with radius of convergence greater than , then has a power series expansion whose coefficients asymptotically approach . Substituting , we conclude that the coefficients of

asymptotically approach

.

But the former is precisely the joint moment generating function of , and the latter is precisely the joint moment generating function of independent Poisson random variables with parameters . The conclusion follows.

Mean and variance

An exact result which can be deduced using the above methods is that the expected number of -cycles of a random permutation of elements is if and otherwise. It follows that the total expected number of cycles is the harmonic number

.

Since Poisson random variables have the same mean and variance, and by the asymptotic independence statement above, we expect the variance of the total number of cycles to also be asymptotic to . This is in fact true, and can be shown using the exponential formula as above.

In The Anatomy of Integers and Permutations, Granville suggested that the decomposition of a random permutation into cycles should be thought of as analogous to the decomposition of a random integer into prime factors. In light of this analogy, the above computation should be thought of as roughly analogous to the Hardy-Ramanujan theorem.

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.

Axiomatics

For convenience, in this section we will restrict to essentially finite groupoids, namely those groupoids equivalent to groupoids with finitely many objects and morphisms.

Associated to any essentially finite groupoid is a rational number, its groupoid cardinality , which is uniquely determined by the following four properties, analogous to the properties uniquely specifying Euler characteristic:

1. Cardinality: , where is the groupoid with one object and one morphism.
2. Homotopy invariance: If ( is equivalent to ), then .
3. Gluing: .
4. Covering: If is an -sheeted covering map, then .

A covering map of groupoids is a functor which is surjective on objects and which satisfies the unique path lifting property: if is a morphism in and is an object in such that , then there exists a unique morphism in such that . This axiomatizes the path lifting property satisfied by a covering map of topological spaces. A covering map is -sheeted if the preimage of every object in consists of objects in .

The homotopy invariance and gluing axioms imply that groupoid cardinality is completely determined by how it behaves on one-object groupoids , where is a finite group (since we are assuming essential finiteness). Associated to any such groupoid is a canonical -sheeted cover

where is the action groupoid for the action of on itself (the objects are the elements of and there is a unique morphism between any pair of objects). This covering map sends the morphism to the element of . The notation is by strong analogy with the theory of classifying spaces.

Since is equivalent to a point, by the cardinality axiom, and the covering axiom then implies that . In conclusion, we find that if is an essentially finite groupoid then, writing the skeleton of as

we have

.

In words, the groupoid cardinality of is a weighted sum over the isomorphism classes of objects in , where an object is weighted by the size of its automorphism group. Intuitively speaking, we can think of the objects of as being “cut up” by their automorphism groups into fractional points.

Groupoid cardinality has other properties besides the above that make it a natural measure of the size of a groupoid.

Proposition: Let be essentially finite groupoids. Then their product is also essentially finite, and .

Proof. A groupoid is essentially finite if and only if it has finitely many isomorphism classes and the objects in each isomorphism class have finitely many automorphisms. This condition is preserved under finite products; moreover, if

and

then

which gives the desired result.

Alternatively, one could show that satisfied all of the axioms above.

Proposition: Let be a finite set and be a finite group acting on . Then the groupoid cardinality of the action groupoid or weak quotient is .

Note that this is badly false for the set-theoretic quotient , a point which trips up many beginners in combinatorics.

The idea of the proof is that we would like to apply the covering axiom to the natural map (thinking of as a discrete groupoid), except that this map isn’t a covering map unless the action of is free. However, it can be replaced by a covering map up to equivalence (a kind of fibrant replacement) essentially using the Borel construction.

Proof. Instead of considering , consider the equivalent groupoid , which consists of pairs where , and where there is a unique morphism for every . Since acts on both and , it acts on this product, and so we can consider the action groupoid and the corresponding map

.

Since acts freely on , this map is a -sheeted covering map. Moreover, and . We can now apply the covering axiom, and the conclusion follows.

For a more pedestrian proof, observe that it suffices by the gluing axiom to prove the statement in the case that the action of on is transitive, where it reduces to the orbit-stabilizer theorem.

Digression: random finite sets

The definition of groupoid cardinality can be extended to tame groupoids, namely those groupoids such that the sum

converges. For any such groupoid, there is a natural probability measure on given by the condition that a given isomorphism class occurs with probability

.

For example, if is the groupoid of finite sets and bijections, then

and the finite set of cardinality occurs with probability . In other words, “size of a random finite set” is Poisson with parameter . It is unclear to me what the significance of this observation is, if any.

More generally, let be a finite set and consider the groupoid of -colored finite sets. This is the groupoid whose objects are finite sets equipped with a map (assigning to each element of its color) and whose morphisms are bijections compatible with colors. The cardinality of this groupoid may be computed in two ways. On the one hand, there are isomorphism types of objects where , and the groupoid consisting these isomorphism types is equivalent to the action groupoid of acting on the set of all functions from an -element set to , hence the groupoid cardinality is

.

On the other hand, the groupoid of -colored finite sets is equivalent to the product of copies of the groupoid of finite sets; the equivalence is given by sending an -colored finite set to the finite sets given by the elements of each color. It is not hard to show that for tame groupoids we have , hence the groupoid cardinality is

.

Hence “size of a random -colored finite set” is Poisson with parameter , and along the way to seeing this we have shown that two ways of defining give the same answer (and also implicitly given a combinatorial proof that ).

There is much more to say about these kinds of arguments, much of which has been said by John Baez at some point, but I don’t know a place where all of the relevant links have been collected. One place to start and work backwards from is week300.

Groupoid cardinality and Euler characteristic

The axiomatic definition of groupoid cardinality suggests that it ought to behave like Euler characteristic, except that the Euler characteristic of familiar spaces are integers and groupoid cardinality is not an integer. However, there is a nice sense in which the Euler characteristic of ought to be .

is a groupoid model of a classifying space of , also denoted , which for discrete groups has two equivalent definitions. It is the unique (up to homotopy) connected space such that and such that all higher homotopy groups are trivial; in other words, it is the Eilenberg-MacLane space . Such spaces are also known as aspherical spaces.

The classifying space is also the space which represents, in a suitable homotopy category, the functor sending a topological space to its set of principal -bundles. When is a discrete group, this is the same thing as a -cover, but the definition in terms of bundles also generalizes to topological groups.

Example. is the circle .

Example. More generally, a nice connected space is a for if and only if its universal cover is contractible; in particular any hyperbolic manifold has this property.

Example. is infinite real projective space .

The sense in which ought to be for finite is the following. Recall that if is, say, a finite CW complex, we should have

where is the number of -cells of . There is a distinguished model of (the space) having a cell decomposition in which , and thus we ought to have

by summing a divergent geometric series! I learned this from MO. This can be seen more explicitly for , for example, which has a single cell in each dimension and therefore whose Euler characteristic ought to be Grandi’s series

.