Differentiable stack

In differential geometry, a differentiable stack is a stack over the category of differentiable manifolds (with the usual open covering topology) which admits an atlas. In other words, a differentiable stack is a stack that can be represented by a Lie groupoid.

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 $\Omega_X^p$ of differential p-forms over X is given by, for any x in X over a manifold U, letting $\Omega_X^p(x)$ be the space of p-forms on U. The sheaf $\Omega_X^0$ is called the structure sheaf on X and is denoted by $\mathcal{O}_X$ . $\Omega_X^*$ 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 $G \to X$ is called a gerbe over X if $G \to G \times_X G$ is also an epimorphism. For example, if X is a stack, $BS^1 \times X \to X$ is a gerbe. A theorem of Giraud says that $H^2(X, S^1)$ corresponds one-to-one to the set of gerbes over X that are locally isomorphic to $BS^1 \times X \to X$ and that come with trivializations of their bands.