Exercise 4, § 3

One defines the mass $\mu(G)$ of a groupoid G as the supremum of the sums $\sum_{a\in A}$ Card(G$_a$)$^{-1}$, where A runs through the set of subsets of Som(G) which meet each orbit of G in at most one point. One says that G is of finite mass if one has $\mu(G)\in \mathbf{R}$; one says that G is essentially finite if the set of its orbits is finite and if the isotropy group at each of its points is finite.

a) Exercise prove that an essentially finite groupoid is of finite mass.

b) Let G be the groupoid associated with a finite group Γ. It is essentially finite and one has $\mu(G) = 1/$ Card(Γ).

c) Let Γ be a finite group, let X be a finite set endowed with a right operation of Γ and let G be the associated groupoid. It is essentially finite and one has $\mu(G) =$ Card(X)$/$ Card(Γ).

d) Let $(G_i)_{i\in I}$ be a family of groupoids of finite mass. The sum groupoid and the product groupoid of the family $(G_i)$ have for mass $\sum\mu(G_i)$ and $\prod\mu(G_i)$ respectively.

e) Let X be a set. One defines a groupoid G whose set of vertices is the set of finite subsets of X and such that, for every pair $(Y_1,Y_2)$ of finite subsets of X, the set of arrows with origin $Y_1$ and term $Y_2$ is the set of bijections of $Y_1$ onto $Y_2$, the composition of the arrows being given by the composition of the mappings. Now prove that G is essentially finite and calculate its mass.