Timeline for Based loop groups as stacks?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Feb 25, 2015 at 10:30 | history | edited | Oliver | CC BY-SA 3.0 |
added 3 characters in body
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| Feb 25, 2015 at 10:29 | comment | added | Oliver | @QiaochuYuan: My understanding is that the dimension of the differentiable stack $[M/G]$, where $G$ is a Lie group acting smoothly on a manifold $M$, is $\dim M - \dim G$. And $BG$ is $[pt/G]$. More generally, the dimension of an arbitrary differentiable stack is computed from any Lie groupoid presentation of it (webusers.imj-prg.fr/~gregory.ginot/papers/DiffStacksIGG2013.pdf). I'm afraid I do not know much beyond the definition. | |
| Feb 25, 2015 at 10:22 | vote | accept | Oliver | ||
| Feb 24, 2015 at 21:22 | comment | added | Elden Elmanto | oh man I really wanted to say $G$ instead of $BG$! | |
| Feb 24, 2015 at 21:01 | answer | added | David Carchedi | timeline score: 10 | |
| Feb 24, 2015 at 20:52 | comment | added | David Carchedi | @EldenElmanto: The homotopy pullback to which you are referring actually gives you $G$ back: $pt \to BG$ is a principal $G$-bundle in the world of stacks, and the pullback becomes a proncipal $G$-bundle over $pt$- and there's only one of these I know about ;-) | |
| Feb 24, 2015 at 19:05 | comment | added | Qiaochu Yuan | In what useful sense does $BG$ have dimension $- \dim (G)$? This is certainly not its cohomological dimension, for example. Does "dimension" here mean something like "Euler characteristic of the tangent complex"? | |
| Feb 24, 2015 at 16:38 | comment | added | Elden Elmanto | Perhaps (I say this because I am not sure of what adjectives (like derived) to put everywhere) you can view it as a homotopy pullback (in an appropriate category of stacks) of the diagram $pt \rightarrow BG \leftarrow pt$. In this way you can calculate its tangent space (or, more appropriately, its cotangent complex) using the sequences induced by being a fibration. | |
| Feb 24, 2015 at 16:25 | answer | added | Nerses Aramian | timeline score: 5 | |
| Feb 24, 2015 at 16:18 | review | First posts | |||
| Feb 24, 2015 at 17:23 | |||||
| Feb 24, 2015 at 16:16 | history | asked | Oliver | CC BY-SA 3.0 |