Connection with Lie groupoids
Every Lie groupoid Γ gives rise to a differentiable stack that is the category of Γ-torsors. In fact, every differentiable stack is of this form. Hence, roughly, "a differentiable stack is a Lie groupoid up to Morita equivalence."
A differentiable space is a differentiable stack with trivial stabilizers. For example, if a Lie group acts freely but not necessarily properly on a manifold, then the quotient by it is in general not a manifold but a differentiable space.
With Grothendieck topology
A differentiable stack X may be equipped with Grothendieck topology in a certain way (see the reference). This gives the notion of a sheaf over X. For example, the sheaf of differential p-forms over X is given by, for any x in X over a manifold U, letting be the space of p-forms on U. The sheaf is called the structure sheaf on X and is denoted by . comes with exterior derivative and thus is a complex of sheaves of vector spaces over X: one thus has the notion of de Rham cohomology of X.
An epimorphism between differentiable stacks is called a gerbe over X if is also an epimorphism. For example, if X is a stack, is a gerbe. A theorem of Giraud says that corresponds one-to-one to the set of gerbes over X that are locally isomorphic to and that come with trivializations of their bands.
- Kai Behrend, Ping Xu, Differentiable Stacks and Gerbes, 2008
- Eugene Lerman, Anton Malkin, Differential characters as stacks and prequantization, 2008