Differentiable stack

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

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.

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, obviously, 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 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.

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