derived smooth geometry
This means that a differentiable stack is in a way nothing else but a Lie groupoid: but the point is that equivalence of stacks captures the notion of Morita equivalence of their presenting (Lie) groupoids.
This means that looking at a Lie groupoid in terms of the stack it presents provides a direct way to recognizing the right notion of equivalence of these objects. The notion of Morita equivalence, on the other hand, proceeds via the reasoning of homotopy theory.
Notice that the stack presented by a (Lie) groupoid is really the stack which sends every test manifold to the category of -principal bundles over that manifold. Such a -principal bundle, also known as a torsor, is analogous to a module for an algebra. This explains the terminology of “Morita morphisms”, which originates in algebra:
Just as two algebras are Morita equivalent if their categories of modules are equivalent, two groupoids are Morita equivalent if their stacks of torsors are equivalent.
Sending a Lie groupoid to the smooth stack it represents constitutes an equivalence of (infinity,1)-categories between Lie groupoids with Morita morphisms/bibundles between them and differentiable stacks.
For , two differentiable stacks, there is the mapping stack , which a priory is only a smooth stack. It should be true that a sufficient condition for this itself to be a Frechet-differentiable stack is that has an atlas by a compact smooth manifold.
(See also manifold structure of mapping spaces.)
David Carchedi, Categorical Properties of Topological and Diffentiable Stacks, PhD thesis, Universiteit Utrecht, 2011
Joost Nuiten, Cohomological quantization of local prequantum boundary field theory, MSc thesis, Utrecht, August 2013