Timeline for Do we have classification (upto Morita equivalence) of Lie groupoids?
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21 events
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| Sep 27, 2020 at 16:41 | comment | added | Praphulla Koushik | @DavidRoberts Yes. I think the same. I will leave it as it is and edit if I can get some interesting special case as a question. | |
| Sep 26, 2020 at 10:28 | comment | added | David Roberts♦ | @PraphullaKoushik yes, it's open ended. Maybe the best you can ask for is for more functors that produce Lie groupoids from other data that end up being not Morita equivalent to the ones on the list. As it is, your list has reasonable overlap (1. and 2. are both special cases of 3., for example; and there's a relation between 4. and 5., since a submersion defines a foliation of its domain, 6. and 7. are special cases, as I mentioned, of the groupoid arising from a rather general fibre bundle...) | |
| Sep 26, 2020 at 8:21 | comment | added | Praphulla Koushik | @DavidRoberts I could not make it any better than "what an arbitrary (good enough) Lie groupoid looks like?"... Do you have any "comment" for a special case or variant of this question? As it is, it looks open ended... :( | |
| Sep 26, 2020 at 8:21 | comment | added | Praphulla Koushik | @DmitriPavlov I could not make it any better than "what an arbitrary (good enough) Lie groupoid looks like?"... Do you have any "comment" for a special case or variant of this question? As it is, it looks open ended... :( | |
| Sep 22, 2020 at 15:48 | comment | added | Praphulla Koushik | @DmitriPavlov I did not doubt when you said the question is confusing.. :) I do think it is slightly unclear.. I will make it clearer.. | |
| Sep 22, 2020 at 15:46 | comment | added | Praphulla Koushik | @DavidRoberts "But who knows what an arbitrary Lie groupoid looks like?" :) I would like to know about this :D.. I will try to make the question clearer.. | |
| Sep 22, 2020 at 1:21 | comment | added | Dmitri Pavlov | @PraphullaKoushik: It appears that I am not alone in being confused as to the actual meaning of your question; David Roberts voiced similar concerns. Maybe you could give us some existing example of “classification into types” that you find acceptable. | |
| Sep 21, 2020 at 10:00 | comment | added | David Roberts♦ | I don't know. The gauge groupoid one can be replaced by the same construction for an arbitrary fibre bundle, and there is the fundamental groupoid of a manifold (also has a Lie groupoid structure), but I think this might be covered by the foliation groupoid example, I can't remember precisely. I mean, all of the examples you give are essentially functorial constructions of Lie groupoids from other data. But who knows what an arbitrary Lie groupoid looks like? There is also the example of bundles of Lie groups, that's not on your list. Really you can only ask for more examples... | |
| Sep 21, 2020 at 8:22 | comment | added | Praphulla Koushik | @DavidRoberts oh, ok. Do you think the question is not well phrased? If anything is not clear, I would like to make it clearer.. | |
| Sep 21, 2020 at 6:20 | comment | added | David Roberts♦ | Well, up to Morita equivalence, every proper Lie groupoid is glued together out of orthogonal actions of compact Lie groups on open unit balls. You don't need étaleness. I was thinking of gerbes not up to stable iso/Morita equivalence, so I guess I take that one back. Every U(1)-gerbe is, up to stable iso, the pullback of the universal gerbe which can be written down as a lifting gerbe (which looks like a proper action of an infinite-dimensional Lie group). | |
| Sep 21, 2020 at 4:22 | comment | added | Praphulla Koushik | @DmitriPavlov I am not asking for interesting examples or constructions of Lie groupoids.. I do not know why it is conveying in that way :O I do not know what other term to use other than classification for what I have mentioned before... Let me rephrase it again... I take all proper etale Lie groupoids to be of "same type" as all of them are locally action Lie groupoids... | |
| Sep 21, 2020 at 4:19 | comment | added | Praphulla Koushik | @DavidRoberts "It will look like none of these, in general".. not even locally? Moerdijk assumes the Lie groupoid is proper and etale to say it is locally an action Lie groupoid.. I saw this in arxiv.org/abs/math/0203100 (page 8).. May be it is true with out etale assumption.. | |
| Sep 21, 2020 at 4:09 | comment | added | David Roberts♦ | Given an appropriate Čech cocycle, one can construct a bundle gerbe from it à la Hitchin–Chatterjee. It will look like none of these, in general. | |
| Sep 21, 2020 at 4:07 | comment | added | David Roberts♦ | You don't need étaleness to get locally like an action groupoid: proper is enough. | |
| Sep 21, 2020 at 3:47 | comment | added | Dmitri Pavlov | So it appears that the actual question being asked here is “What are some interesting examples or constructions of Lie groupoids?”. This is radically different from classifying Lie groupoids in any sense, since a typical example of a classification in this area is the Cartan–Killing classification of simple Lie group, a precise and complete result. This newly revealed question, on the other hand, is clearly open-ended and does not have (or allow for) a complete answer. | |
| Sep 21, 2020 at 2:55 | comment | added | Praphulla Koushik | @DmitriPavlov "type" of a Lie groupoid is not a standard terminology, I just invented now... I call Lie groupoids coming from Lie groups as one type.. Lie groupoids coming from manifolds as one type.. Lie groupoids coming from foliations as one type and so on... Let me rephrase this in another way. Are there any "interesting" examples of Lie groupoids which are not Morita equivalent to any of the above 7 types? Here, interesting does not have a unique meaning and I leave it to you to interpret in a fair manner :) | |
| Sep 21, 2020 at 2:44 | comment | added | Dmitri Pavlov | What is a “type” of a Lie groupoid? Your list of 7 items looks like a fairly ordinary list of examples of Lie groupoids that one would give in a typical expository text after a definition. Clearly, there are many more examples, and I do not understand why one would expect to exhaust all Lie groupoids by adding other examples to this list. | |
| Sep 21, 2020 at 2:20 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| Sep 21, 2020 at 2:14 | comment | added | Praphulla Koushik | @DmitriPavlov It looks like I have not mentioned it clearly.. Yes, I do know that there is no hope of classifying Lie groups... I do not want to classify Lie groups :D For me Lie groupoids coming from Lie groups are of "same type"... Does it now make it clear what I was trying to understand? | |
| Sep 20, 2020 at 20:05 | comment | added | Dmitri Pavlov | Lie groups sit inside Lie groupoids via the delooping construction, and there is no hope of classifying Lie groups. | |
| Sep 20, 2020 at 15:50 | history | asked | Praphulla Koushik | CC BY-SA 4.0 |