Mathematical constants and functions - Revision history

en>Ignacitum: Silverman constant

2015-01-06T21:00:16Z

Silverman constant

← Older revision		Revision as of 23:00, 6 January 2015
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en>Ignacitum: Lebesgue constant (interpolation)

2014-03-03T21:18:41Z

Lebesgue constant (interpolation)

← Older revision		Revision as of 23:18, 3 March 2014
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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]

	~~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.~~

	~~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''.~~

	~~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.~~

	~~== References ==~~
	~~{{reflist}}~~
	*Kai Behrend, Ping Xu, [http://~~arxiv~~.~~org/abs/math/0605694 Differentiable Stacks and Gerbes], 2008~~
	*Eugene Lerman, Anton Malkin, [http://arxiv.~~org/pdf~~/~~0710.4340v2.pdf Differential characters as stacks and prequantization], 2008~~

	~~== External links =~~=
	*http://ncatlab.org/nlab/show/~~differentiable+stack~~

	~~[[Category:Differential geometry~~]]


	~~{{geometry-stub}}~~

en>Ignacitum: Plastic number

2014-01-30T00:59:52Z

Plastic number

New page

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>

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.

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''.

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.

== References ==
{{reflist}}
*Kai Behrend, Ping Xu, [http://arxiv.org/abs/math/0605694 Differentiable Stacks and Gerbes], 2008
*Eugene Lerman, Anton Malkin, [http://arxiv.org/pdf/0710.4340v2.pdf Differential characters as stacks and prequantization], 2008

== External links ==
*http://ncatlab.org/nlab/show/differentiable+stack

[[Category:Differential geometry]]

{{geometry-stub}}