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According to "Notes on differentiable stacks" by Heinloth,

the classifying stack will also classify $G$-bundles on stacks. (Remark 2.13)

(Here $G$ is a Lie group.) My questions are:

(1) What is the precise statement? (The category of morphisms to the classifying stack $[pt/G]$ is equivalent to the category of principal $G$-bundles? If so, how do we define morphisms between principal $G$-bundles?)

(2) How do we prove the statement?

(3) Suppose we have a principal G-bundle over a differentiable stack $\mathcal{X}$. How do we construct a morphism from $\mathcal{X}$ to the classifying stack?

I would be most grateful if you could tell me any good references.

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1 Answer 1

up vote 2 down vote accepted

They key insight is that the bicategory of differentiable stacks is equivalent to the bicategory of Lie groupoids, where the 1-morphisms are so-called bibundles, or Hilsum-Skandalis morphisms. Under this equivalence, the statement is the following:

Let $\mathcal{X}$ be a Lie groupoid, and let $BG$ the Lie groupoid that represents the stack $[pt/G]$. Then there is an equivalence of categories $$ Bun_G(\mathcal{X}) = Hom_{LieGrpd}(\mathcal{X},BG). $$

In fact, this equivalence is now tautological, thanks to the translation into the Lie groupoid world.

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