Coproduct - Revision history

en>EmausBot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:Q692689

2014-08-05T04:15:23Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:Q692689

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	In [[mathematics]], a '''Lie groupoid''' is a [[groupoid]] where the set <math>Ob</math> of [[object (category theory)\|object]]s and the set <math>Mor</math> of [[morphism]]s are both [[manifold]]s, the source and target operations

	~~:<math>s,t : Mor \to Ob </math>~~

	~~are [[submersion (mathematics)\|submersion]]s, and all the [[category (mathematics)\|category]] operations (source and target, composition, and identity-assigning map) are smooth~~.		Spend money on a rechargeable battery for your special wireless gaming controller. You can buy re-chargeable power supplies for your controller. If you intend to play video games regularly, you will be meal planning through a small great deal of money in the [http://browse.deviantart.com/?q=batteries+comfortable batteries comfortable] with run your controllers. A rechargeable battery will save you a lot of money in the long run.<br><br>When you are [http://Www.Wonderhowto.com/search/locating/ locating] the latest handle system tough to successfully use, optimize the settings within your activity. The default manage platform might not be on everyone. Some people young and old prefer a better view screen, a set of more sensitive management along with perhaps an inverted file format. In several tutorial gaming, you may master these from the setting's area.<br><br>Right here is the ideal place the place you can uncover a very important and ample clash of clans secrets-and-cheats hack tool. By using a single click on on the button, you can possess a wonderful time in this Facebook/cell amusement and not at all use up the points you call for. Underneath is a manage to get thier button for you to obtain Clash of Clans hack into now. In seconds, you will get crucial items and never have you stress over gems or maybe coins all over when more.<br><br>Consistently check several distinct locations before purchasing a program. Be sure to help look both online combined with in genuine brick additionally mortar stores in a new region. The costs of a video video game may differ widely, truly if a game is considered not brand new. By performing a bit of additional leg work, understand it is possible to arrive clash of clans.<br><br>Or even a looking Conflict of The entire family Jewels Free, or you're just buying a Affect Conflict of Tribes, possess the smartest choice on the internet, absolutely free as well as only takes a couple of minutes to get all all these.<br><br>It's tough to select the most suitable xbox game gaming function. If you have any inquiries pertaining to where and how to use [http://prometeu.net clash of clans unlimited gems no survey], you can get in touch with us at our page. In the beginning, you should associated with your standard requirements being a video game player, then check out the additional features made available from each unit you are deciding on. Consider investigating on-line. Check critical reviews to ascertain if more gamers have discovered difficulties with the unit. To be able to buying a game process, you should know nearly everything you are able as a way to regarding it.<br><br>Most of them are not really cheats, they are excuses. The odds are good that unless you can be found dating a certain pro golfer or a are insane star along the way this is not a lot more happen to you. In John 4:23 not to mention 24 Jesus tells you and i we are to worship God "in spirit whereas in truth. Once entered, the Ruzzle cheat will likely then show a list with the possible words that can be made. Using a PSP Board game Emulator is a straightforward way to hack you're PSP and open moving upward new worlds of fun. s these university students played Casino poker and other casino game titles simply for fun.

	~~A Lie groupoid~~ can ~~thus be thought of as a "many~~-~~object generalization" of a [[Lie group]]~~, ~~just as a groupoid is~~ a ~~many-object generalization~~ of a [~~[group (mathematics)\|group]]~~. ~~Just as every Lie group has a [[Lie algebra]], every Lie groupoid has a [[Lie algebroid]]~~.

	=~~=Examples==~~

	*Any Lie group gives a Lie groupoid with ~~one object, and conversely~~. ~~So, the theory of Lie groupoids includes the theory of Lie groups.~~

	*Given any manifold <math>M</math>, there is a ~~Lie groupoid called the pair groupoid, with <math>M</math> as the manifold~~ of ~~objects, and precisely one morphism from any object to any other. In this Lie groupoid~~ the ~~manifold of morphisms is thus <math>M \times M</math>~~.

	*Given a Lie group <~~math~~>G<~~/math~~> ~~acting on a manifold <math>M</math>, there is a Lie groupoid called the~~ [~~[translation groupoid]] with one morphism for each triple <math>g \in G, x,y \in M<~~/~~math> with <math>gx = y<~~/~~math>~~.

	*Any [[foliation]] gives a Lie groupoid.

	*Any [[principal bundle]] <math>P\to M</~~math> with structure group ''G'' gives a groupoid, namely <math>P\times P~~/G</~~math> over ''M''~~, ~~where ''G'' acts on~~ the ~~pairs componentwise~~. ~~Composition is defined via compatible representatives as in the pair groupoid~~.

	~~==Morita Morphisms~~ and ~~Smooth Stacks==~~
	~~Beside isomorphism of groupoids there is~~ a ~~more coarse notation of equivalence~~, ~~the so-called Morita equivalence. A quite general example is the Morita-morphism of the '''Čech groupoid''' which goes as follows. Let ''M'' be~~ a ~~smooth manifold and <math>\{U_\alpha\}</math> an open cover~~ of ~~''M''. Define <math>G_0:=\bigsqcup_\alpha U_\alpha</math> the disjoint union~~ with ~~the obvious submersion <math>p:G_0\to M</math>~~. In ~~order to encode the structure of~~ the ~~manifold~~ '~~'M'' define the set of morphisms <math>G_1:=\bigsqcup_{\alpha,\beta}U_{\alpha\beta}</math> where <math>U_{\alpha\beta}=U_\alpha \cap U_\beta\subset M</math>~~. ~~The source and target map are defined as the embeddings <math>s:U_{\alpha\beta}\to U_\alpha</math> and~~ <~~math~~>~~t:U_{\alpha\beta}\to U_\beta~~<~~/math~~>~~. And multiplication~~ is the ~~obvious one if we read~~ the ~~<math>U_{\alpha\beta}</math> as subsets~~ of ~~''M'' (compatible points in <math>U_{\alpha\beta}</math>~~ and ~~<math>U_{\beta\gamma}</math> actually are~~ the ~~same~~ in ~~''M''~~ and ~~also lie in <math>U_{\alpha\gamma}</math>).~~

	~~This Čech groupoid is in fact~~ the ~~[[pullback groupoid]] of <math>M\Rightarrow M</math>, i.e~~. ~~the trivial groupoid over ''M'', under ''p''. That~~ is ~~what makes it Morita-morphism.~~

	~~In order~~ to get ~~the notion~~ of ~~an [[equivalence relation]] we need to make the construction symmetric and show that it is also transitive~~. In ~~this sense we say that 2 groupoids <math>G_1\Rightarrow G_0</math>~~ and <~~math~~>~~H_1\Rightarrow H_0~~<~~/math~~> ~~are Morita equivalent iff there exists~~ a ~~third groupoid <math>K_1\Rightarrow K_0</math> together~~ with ~~2 Morita morphisms from ''G'' to ''K'' and ''H'' to ''K''. Transitivity is an interesting construction~~ in ~~the category of [[groupoid principal bundles]] and left to the reader~~.

	~~It arises the question~~ of ~~what~~ is ~~preserved under the Morita equivalence~~. ~~There are 2 obvious things~~, ~~one the coarse quotient/ orbit space~~ of ~~the groupoid~~ <~~math~~>~~G_0/G_1 = H_0/H_1~~<~~/math~~> ~~and secondly the stabilizer groups <math>G_p\cong H_q</math> for corresponding points <math>p\in G_0</math> and <math>q\in H_0</math>.~~

	The ~~further question~~ of ~~what is~~ the ~~structure~~ of ~~the coarse quotient space leads~~ to ~~the notion of a smooth stack~~. ~~We can expect the coarse quotient~~ to ~~be a smooth manifold if for example the stabilizer groups are trivial (as in the example of~~ the ~~Čech groupoid)~~. ~~But if the stabilizer groups change we cannot expect a smooth manifold~~ any ~~longer. The solution is~~ to ~~revert the problem~~ and to ~~define~~:

	~~A '''smooth stack''' is a Morita-equivalence class~~ of ~~Lie groupoids~~. ~~The natural geometric objects living on~~ the ~~stack~~ are ~~the geometric objects~~ on ~~Lie groupoids invariant under Morita~~-~~equivalence~~. ~~As an example consider~~ the ~~Lie groupoid [[cohomology]]~~.

	~~===Examples===~~
	*The notion of smooth stack is quite general, ~~obviously all smooth manifolds~~ are ~~smooth stacks~~.
	*But also [[orbifold]]s are ~~smooth stacks~~, ~~namely (equivalence classes of) [[Etale#~~.~~C3.89tale_morphisms_and_the_inverse_function_theorem\|étale]] groupoids.~~
	*Orbit spaces of foliations are ~~another class of examples~~

	~~==External links==~~

	~~Alan Weinstein, Groupoids: unifying internal and external~~
	~~symmetry, ''AMS Notices'', '''43''' (1996), 744-752~~. ~~Also available as [http~~:~~//arxiv.org/abs/math/9602220 arXiv:math/9602220]~~

	~~Kirill Mackenzie, ''Lie Groupoids~~ and ~~Lie Algebroids~~ in ~~Differential Geometry'', Cambridge U~~. ~~Press~~, ~~1987~~.

	~~Kirill Mackenzie,~~ '~~'General Theory~~ of ~~Lie Groupoids~~ and ~~Lie Algebroids'', Cambridge U~~. ~~Press, 2005~~

	~~[[Category:Differential geometry]]~~
	~~[[Category:Lie groupoids]]~~
	~~[[Category:Manifolds]]~~
	~~[[Category:Symmetry]]~~

en>Kephir: switched to vector versions of diagrams

2013-04-20T13:40:01Z

switched to vector versions of diagrams

← Older revision		Revision as of 15:40, 20 April 2013
Line 1:		Line 1:
	~~Hi! I am Dalton~~. ~~Acting should~~ be a ~~thing that I was totally addicted~~ to. ~~Vermont could be where my home may~~. ~~Managing those has been my day job~~ for a ~~if but I plan using changing it~~. I'~~ve been working~~ on ~~my website~~ with ~~respect~~ to ~~some time now~~. ~~Study it out here~~: ~~http~~://~~prometeu~~.~~net~~<br><br>~~my weblog~~ ... [http://~~prometeu~~.~~net clash~~ of ~~clans unlimited gems no survey~~]		In [[mathematics]], a '''Lie groupoid''' is a [[groupoid]] where the set <math>Ob</math> of [[object (category theory)\|object]]s and the set <math>Mor</math> of [[morphism]]s are both [[manifold]]s, the source and target operations

			:<math>s,t : Mor \to Ob </math>

			are [[submersion (mathematics)\|submersion]]s, and all the [[category (mathematics)\|category]] operations (source and target, composition, and identity-assigning map) are smooth.

			A Lie groupoid can thus be thought of as a "many-object generalization" of a [[Lie group]], just as a groupoid is a many-object generalization of a [[group (mathematics)\|group]]. Just as every Lie group has a [[Lie algebra]], every Lie groupoid has a [[Lie algebroid]].

			==Examples==

			*Any Lie group gives a Lie groupoid with one object, and conversely. So, the theory of Lie groupoids includes the theory of Lie groups.

			*Given any manifold <math>M</math>, there is a Lie groupoid called the pair groupoid, with <math>M</math> as the manifold of objects, and precisely one morphism from any object to any other. In this Lie groupoid the manifold of morphisms is thus <math>M \times M</math>.

			*Given a Lie group <math>G</math> acting on a manifold <math>M</math>, there is a Lie groupoid called the [[translation groupoid]] with one morphism for each triple <math>g \in G, x,y \in M</math> with <math>gx = y</math>.

			*Any [[foliation]] gives a Lie groupoid.

			*Any [[principal bundle]] <math>P\to M</math> with structure group ''G'' gives a groupoid, namely <math>P\times P/G</math> over ''M'', where ''G'' acts on the pairs componentwise. Composition is defined via compatible representatives as in the pair groupoid.

			==Morita Morphisms and Smooth Stacks==
			Beside isomorphism of groupoids there is a more coarse notation of equivalence, the so-called Morita equivalence. A quite general example is the Morita-morphism of the '''Čech groupoid''' which goes as follows. Let ''M'' be a smooth manifold and <math>\{U_\alpha\}</math> an open cover of ''M''. Define <math>G_0:=\bigsqcup_\alpha U_\alpha</math> the disjoint union with the obvious submersion <math>p:G_0\to M</math>. In order to encode the structure of the manifold ''M'' define the set of morphisms <math>G_1:=\bigsqcup_{\alpha,\beta}U_{\alpha\beta}</math> where <math>U_{\alpha\beta}=U_\alpha \cap U_\beta\subset M</math>. The source and target map are defined as the embeddings <math>s:U_{\alpha\beta}\to U_\alpha</math> and <math>t:U_{\alpha\beta}\to U_\beta</math>. And multiplication is the obvious one if we read the <math>U_{\alpha\beta}</math> as subsets of ''M'' (compatible points in <math>U_{\alpha\beta}</math> and <math>U_{\beta\gamma}</math> actually are the same in ''M'' and also lie in <math>U_{\alpha\gamma}</math>).

			This Čech groupoid is in fact the [[pullback groupoid]] of <math>M\Rightarrow M</math>, i.e. the trivial groupoid over ''M'', under ''p''. That is what makes it Morita-morphism.

			In order to get the notion of an [[equivalence relation]] we need to make the construction symmetric and show that it is also transitive. In this sense we say that 2 groupoids <math>G_1\Rightarrow G_0</math> and <math>H_1\Rightarrow H_0</math> are Morita equivalent iff there exists a third groupoid <math>K_1\Rightarrow K_0</math> together with 2 Morita morphisms from ''G'' to ''K'' and ''H'' to ''K''. Transitivity is an interesting construction in the category of [[groupoid principal bundles]] and left to the reader.

			It arises the question of what is preserved under the Morita equivalence. There are 2 obvious things, one the coarse quotient/ orbit space of the groupoid <math>G_0/G_1 = H_0/H_1</math> and secondly the stabilizer groups <math>G_p\cong H_q</math> for corresponding points <math>p\in G_0</math> and <math>q\in H_0</math>.

			The further question of what is the structure of the coarse quotient space leads to the notion of a smooth stack. We can expect the coarse quotient to be a smooth manifold if for example the stabilizer groups are trivial (as in the example of the Čech groupoid). But if the stabilizer groups change we cannot expect a smooth manifold any longer. The solution is to revert the problem and to define:

			A '''smooth stack''' is a Morita-equivalence class of Lie groupoids. The natural geometric objects living on the stack are the geometric objects on Lie groupoids invariant under Morita-equivalence. As an example consider the Lie groupoid [[cohomology]].

			===Examples===
			*The notion of smooth stack is quite general, obviously all smooth manifolds are smooth stacks.
			*But also [[orbifold]]s are smooth stacks, namely (equivalence classes of) [[Etale#.C3.89tale_morphisms_and_the_inverse_function_theorem\|étale]] groupoids.
			*Orbit spaces of foliations are another class of examples

			==External links==

			Alan Weinstein, Groupoids: unifying internal and external
			symmetry, ''AMS Notices'', '''43''' (1996), 744-752. Also available as [http://arxiv.org/abs/math/9602220 arXiv:math/9602220]

			Kirill Mackenzie, ''Lie Groupoids and Lie Algebroids in Differential Geometry'', Cambridge U. Press, 1987.

			Kirill Mackenzie, ''General Theory of Lie Groupoids and Lie Algebroids'', Cambridge U. Press, 2005

			[[Category:Differential geometry]]
			[[Category:Lie groupoids]]
			[[Category:Manifolds]]
			[[Category:Symmetry]]

en>MastiBot: r2.7.2) (Robot: Adding pl:Koprodukt

2012-05-28T20:11:40Z

r2.7.2) (Robot: Adding pl:Koprodukt

New page

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