Lie Groupoid and Differentiable Stacks

Roughly speaking, differentiable stacks are Lie groupoids up to Morita equivalence.

• Any Lie groupoid \(X\) defines a differentiable stack \(\mathcal{X}\) of \(X\)-torsors. Two differentiable stacks \(\mathcal{X}\) and \(\mathcal{X}'\) are isomorphic iff the Lie groupoids \(X\) and \(X'\) are Morita equivalent.
• In a certain sense, Lie groupoids are like “local charts” on a differentiable stack.

See Differentiable Stacks and Gerbes for more details, see also Stacky Lie groups for a brief review about how to go back and forward between differentiable stacks and Lie groupoids.