higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
∞-Lie theory (higher geometry)
A differentiable stack is a geometric stack on the site SmoothMfd of smooth manifolds: a stack which is represented by a Lie groupoid.
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 $G$ is really the stack which sends every test manifold to the category of $G$-principal bundles over that manifold. Such a $G$-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.
(e.g. Blohmann 07, Carchedi 11). For review in a broader context see also Nuiten 13, around prop. 2.2.34
A differentible stack is in particular an object in the cohesive (∞,1)-topos Smooth∞Grpd that is
It is however more special than that. The general 1-truncated concrete smooth ∞-groupoids are internal groupoids in diffeological spaces.
For $X$, $Y$ two differentiable stacks, there is the mapping stack $[X,Y]$, 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 $X$ has an atlas by a compact smooth manifold.
(See also manifold structure of mapping spaces.)
differentiable stack
Christian Blohmann, Stacky Lie groups, Int. Mat. Res. Not. (2008) Vol. 2008: article ID rnn082 (arXiv:math/0702399)
Kai Behrend, Ping Xu, Differentiable Stacks and Gerbes J. Symplectic Geom. Volume 9, Number 3 (2011), 285-341. (arXiv:math/0605694)
Jochen Heinloth, Some notes on differentiable stacks (pdf)
Richard Hepworth, Vector fields and flows on differentiable stacks (arXiv).
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
See also the references at geometric stack and topological stack.