# Differentiable stack

In differential geometry, a **differentiable stack** is a stack over the category of differentiable manifolds (with the usual open covering topology).

## 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."^{[1]}

## Differential space

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*.

## Gerbes

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.

## References

- Kai Behrend, Ping Xu, Differentiable Stacks and Gerbes, 2008
- Eugene Lerman, Anton Malkin, Differential characters as stacks and prequantization, 2008