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.
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:
- Cardinality: , where is the groupoid with one object and one morphism.
- Homotopy invariance: If ( is equivalent to ), then .
- Gluing: .
- 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
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
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