Differentiable stack

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 ${\displaystyle \Omega _{X}^{p}}$ of differential p-forms over X is given by, for any x in X over a manifold U, letting ${\displaystyle \Omega _{X}^{p}(x)}$ be the space of p-forms on U. The sheaf ${\displaystyle \Omega _{X}^{0}}$ is called the structure sheaf on X and is denoted by ${\displaystyle {\mathcal {O}}_{X}}$. ${\displaystyle \Omega _{X}^{*}}$ 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.

An epimorphism between differentiable stacks ${\displaystyle G\to X}$ is called a gerbe over X if ${\displaystyle G\to G\times _{X}G}$ is also an epimorphism. For example, if X is a stack, ${\displaystyle BS^{1}\times X\to X}$ is a gerbe. A theorem of Giraud says that ${\displaystyle H^{2}(X,S^{1})}$ corresponds one-to-one to the set of gerbes over X that are locally isomorphic to ${\displaystyle BS^{1}\times X\to X}$ and that come with trivializations of their bands.

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

• http://ncatlab.org/nlab/show/differentiable+stack