If $G$ is a finite group, then inside the category of $G$-sets and $G$-maps there is the subcategory whose objects are the orbits (transitive $G$-sets) and whose morphisms are the isomorphisms. I have been calling this groupoid the Weyl groupoid of $G$ and denoting it by $W_G$; it is equivalent to the disjoint union, over subgroups $H$, one in each conjugacy class, of the Weyl group $W_GH$. Is there a standard name and notation for it?
Is there a standard name for functors from $W_G$ to vector spaces? I have been calling them Weyl representations.
I know that the forgetful functor from Mackey functors to Weyl representations induces an isomorphism of Grothendieck groups (even though it is not an equivalence of categories). I would be happy to have a reference for this fact.
