Mathematical constants and functions

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."

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 ${\mathrm{\Omega }}_X^p$ of differential p-forms over X is given by, for any x in X over a manifold U, letting ${\mathrm{\Omega }}_X^p(x)$ be the space of p-forms on U. The sheaf ${\mathrm{\Omega }}_X^0$ is called the structure sheaf on X and is denoted by ${\mathcal{𝒪}}_X$. ${\mathrm{\Omega }}_X^{*}$ 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, $B{S}^1\times X\to X$ 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 $B{S}^1\times X\to X$ and that come with trivializations of their bands.

References

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

External links

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

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