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| In differential geometry, a '''differentiable stack''' is a [[stack (mathematics)|stack]] over the category of [[differentiable manifold]]s (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]]."<ref>{{harvnb|Behrend–Xu|2008}}</ref>
| | 34 year old Wood Machinist Dominic Priestly from Sainte-Catherine, has numerous passions which include becoming a child, property developers in singapore and crossword puzzles. Finished a cruise ship experience that included passing by Prambanan Temple Compounds.<br><br>my web site: [http://www.sierratel.sl/?q=node/16340 http://www.sierratel.sl/?q=node/16340] |
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| A '''differentiable space''' is a differentiable stack with trivial stabilizers. For example, if a [[Lie group]] [[Lie group action|acts]] freely but not necessarily properly on a manifold, then the quotient by it is in general not a manifold but a differentiable space.
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| 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 <math>\Omega_X^p</math> of differential ''p''-forms over ''X'' is given by, for any ''x'' in ''X'' over a manifold ''U'', letting <math>\Omega_X^p(x)</math> be the space of ''p''-forms on ''U''. The sheaf <math>\Omega_X^0</math> is called the [[structure sheaf]] on ''X'' and is denoted by <math>\mathcal{O}_X</math>. <math>\Omega_X^*</math> 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''.
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| An epimorphism between differentiable stacks <math>G \to X</math> is called a [[gerbe]] over ''X'' if <math>G \to G \times_X G</math> is also an epimorphism. For example, if ''X'' is a stack, <math>BS^1 \times X \to X</math> is a gerbe. A theorem of Giraud says that <math>H^2(X, S^1)</math> corresponds one-to-one to the set of gerbes over ''X'' that are locally isomorphic to <math>BS^1 \times X \to X</math> and that come with trivializations of their [[band (geometry)|band]]s.
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| == References ==
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| {{reflist}}
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| *Kai Behrend, Ping Xu, [http://arxiv.org/abs/math/0605694 Differentiable Stacks and Gerbes], 2008
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| *Eugene Lerman, Anton Malkin, [http://arxiv.org/pdf/0710.4340v2.pdf Differential characters as stacks and prequantization], 2008
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| == External links ==
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| *http://ncatlab.org/nlab/show/differentiable+stack
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| [[Category:Differential geometry]]
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| {{geometry-stub}}
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Revision as of 23:18, 3 March 2014
34 year old Wood Machinist Dominic Priestly from Sainte-Catherine, has numerous passions which include becoming a child, property developers in singapore and crossword puzzles. Finished a cruise ship experience that included passing by Prambanan Temple Compounds.
my web site: http://www.sierratel.sl/?q=node/16340