What is Differentiable stack?

A differentiable stack is the analogue in differential geometry of an algebraic stack in algebraic geometry. It can be described either as a stack over differentiable manifolds which admits an atlas, or as a Lie groupoid up to Morita equivalence.

Differentiable stacks are particularly useful to handle spaces with singularities , which appear naturally in differential geometry but are not differentiable manifolds. For instance, differentiable stacks have applications in foliation theory, Poisson geometry and twisted K-theory.