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  • Here the boldface notation is to indicate that this Lie groupoid is a smooth refinement of the classifying space BG???TopB G \in Top of the Lie group: in fact, where BGB G gives isomorphism classes of smooth GG-principal bundles, BG\mathbf{B}G also remembers the isomorphisms themselves ??? and hence in particular the automorphisms ??? of these bundles: it is the moduli stack of smooth GG-principal
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