Timeline for What is the natural Lie groupoid structure on the Atiyah Lie groupoid of a principal $G$-bundle?
Current License: CC BY-SA 4.0
15 events
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| Jul 29, 2020 at 7:21 | history | edited | YCor | CC BY-SA 4.0 |
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| Jul 28, 2020 at 5:05 | vote | accept | Adittya Chaudhuri | ||
| Jul 28, 2020 at 4:57 | comment | added | Adittya Chaudhuri | @DavidRoberts Thank you Sir. | |
| Jul 28, 2020 at 0:17 | answer | added | Dmitri Pavlov | timeline score: 6 | |
| Jul 27, 2020 at 23:42 | comment | added | David Roberts♦ | The objects as defined on the nLab and on Wikipedia give isomorphic sets. That the morphism sets are the same is less obvious, but it boils down to knowing that a map between principal homogeneous $G$-spaces is determined entirely by its value at a single point. It's worth thinking about the case of a trivialisable bundle first. | |
| Jul 27, 2020 at 22:47 | comment | added | Adittya Chaudhuri | @LSpice Ok I got your point. Thanks. | |
| Jul 27, 2020 at 22:38 | comment | added | LSpice | There is no definition in the 'idea' section; it is just an idea, and, as not rigorously defined, cannot be checked against a rigorous definition. | |
| Jul 27, 2020 at 22:33 | comment | added | Adittya Chaudhuri | @LSpice But even if those 2 notions are same I cannot immediately see how they are same. For the definition given in ncatlab.org/nlab/show/… the smooth structure on Objects and morphisms are clear but from here how can I guess the smooth structures for the definition given in the "Idea" section ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea | |
| Jul 27, 2020 at 22:29 | comment | added | LSpice | (Also, you don't sound stupid; quite possibly my reference to Atiyah's paper is stupid. It's just that, if I'm looking for the literature on a concept with someone's name attached to it, I always start by looking at whether that person defined it and, if so, where. I haven't met this concept, so went to Wikipedia to see where Atiyah had defined it.) | |
| Jul 27, 2020 at 22:27 | comment | added | LSpice | Yes, that is what I am saying. | |
| Jul 27, 2020 at 22:26 | comment | added | Adittya Chaudhuri | @LSpice I was asking about the definition given in the section idea ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea not ncatlab.org/nlab/show/…. So are you saying both are actually same notion and the definition given in the idea section is just an informal notion? | |
| Jul 27, 2020 at 22:21 | comment | added | Adittya Chaudhuri | @LSpice Sorry if I sound stupid but I asked about Atiyah Lie Groupoids not Atiyah algebroid. According to wikipedia reference Atiyah Lie Algeroid is the Lie Algebroid of a Gauge Groupoid of a Principal bundle. How from this I can guess the smooth structure on $Obj(At(P))$ and $MorAt(P)$? Can you please explain a bit in little detail? | |
| Jul 27, 2020 at 22:11 | comment | added | LSpice |
At least according to Wikipedia, the obvious place to start with the literature is Atiyah - Complex analytic connections in fibre bundles. Also, your definitions don't seem to match the nLab's, which declares $X$ the objects and $(P \times P)/G$ the morphisms, both with obvious smooth structure. (Iincidentally, note if desired you can replace \lbrace\rbrace by \{\}.)
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| Jul 27, 2020 at 22:10 | history | edited | LSpice | CC BY-SA 4.0 |
\operatorname; deleted 'thanks'
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| Jul 27, 2020 at 21:53 | history | asked | Adittya Chaudhuri | CC BY-SA 4.0 |