nLab
differentiable stack

Definition

A differentiable stack is a stack on the site Diff of manifolds which is presented by a Lie groupoid.

Remarks

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.

Literature

• Kai Behrend, Ping Xu, Differentiable Stacks and Gerbes (arXiv)

• Jochen Heinloth, Some notes on differentiable stacks (pdf)

• Richard A. Hepworth, Vector fields and flows on differentiable stacks (arXiv).