Differentiable stack

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."

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 Ω X p of differential p-forms over X is given by, for any x in X over a manifold U, letting Ω X p ( x ) be the space of p-forms on U. The sheaf Ω X 0 is called the structure sheaf on X and is denoted by O X . Ω 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 → X is called a gerbe over X if G → G × X G is also an epimorphism. For example, if X is a stack, B S 1 × X → 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 B S 1 × X → X and that come with trivializations of their bands.