By R. Switzer
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Extra resources for Algebraic Topology, Homotopy and Homology
G. two ﬁnite groups with the same number of conjugacy classes, hence the same number of isomorphism classes of irreducible representations). We obtain a more geometric notion of Morita equivalence for groups by considering actions on manifolds rather than on linear spaces. e. manifolds where G acts on the left, H acts on the right, and the actions commute. The “tensor product” of such bimodules is deﬁned by the orbit space G XH ∗ H YK := X × Y/H, where H acts on X × Y by (x, y) → (xh, h−1 y). The result of this operation may no longer be smooth, even if X and Y are.
This can also be described as an inﬁnite-dimensional symplectic reduction. The resulting space is a groupoid but may not be a manifold. When it is a manifold, it is the source-simply-connected symplectic groupoid of P . When G(P ) is not a manifold, as the leaf space of a foliation, it can be considered as a diﬀerentiable stack, and even as a symplectic stack. 2 Symplectic groupoids 43 groupoid; we will call it an S-groupoid. The ﬁrst steps of this program have been carried out by Tseng and Zhu .
If one is given two left modules (one could do the same for right modules, of course), one can apply the tensor product construction by changing the “handedness” of one of them. Thus, if S and S are left P modules, then S is a right module, and we can form the tensor product S ∗ S. We call this the classical intertwiner space [95, 96] of S and S and denote it by Hom(S, S ). The name and notation come from the case of modules over an algebra, where the tensor product Y∗ ⊗ X is naturally isomorphic to the space of module homomorphisms from Y to X when these modules are “ﬁnite dimensional”.