Equivalence between the Morita categories of étale Lie groupoids and of locally grouplike Hopf algebroids *

Any étale Lie groupoid G is completely determined by its associated convolution algebra Cc∞(G) equipped with the natural Hopfalgebroid structure. We extend this result to the generalized morphisms between étale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated Cc∞(G)-Cc∞(H)-bimodule Cc∞(P) equipped with the natural coalgebra structure. Furthermore, we prove that the functor Cc∞gives an equivalence between the Morita category of étale Lie groupoids and the Morita category of locally grouplike Hopf algebroids.