Equivalence between the Morita categories of étale Lie groupoids and of locally grouplike Hopf algebroids. (English) Zbl 1181.22008

An étale Lie groupoid \(G\) is completely determined by its convolution algebra \(\mathcal C_c^{\infty }(G)\) equipped with the natural Hopf algebroid structure.
The paper has two principal results: 1) any principal \(H\)-bundle \(P\) over \(G\) is uniquely determined by the associated \(\mathcal C_c^{\infty }(G)-\mathcal C_c^{\infty }(H)\)-bimodule \(\mathcal C_c^{\infty }(P)\) equipped with the natural coalgebra structure, and 2) the functor \(\mathcal C_c^{\infty }\) gives an equivalence between the Morita categories of étale Lie groupoids and of locally grouplike Hopf algebroids.


22A22 Topological groupoids (including differentiable and Lie groupoids)
58H05 Pseudogroups and differentiable groupoids
16T05 Hopf algebras and their applications
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