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It's a well known fact two Lie groupoids are Morita-equivalent iff they induce isomorphic differentiable stacks (I'll call this statement "(1)"). It's also well known that there is a ...

For a Lie group $G$, we can define a principal $G$ bundle as a submersion of manifolds $\pi:P \to X$ equipped with a free right $G$-action on $P$ that is transitive on the fibres over $X$. What goes ...

First of all I should apologies if this question does not count as a research level one. I asked the same question on MathUnderflow and didn't receive any answer. Let me cross post (copy and paste) it ...