Timeline for Stack associated to Lie group and manifold
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 21, 2019 at 16:43 | vote | accept | Praphulla Koushik | ||
| Feb 21, 2019 at 15:56 | comment | added | Dmitri Pavlov | @PraphullaKoushik: The loop-deloop adjunction is discussed in Goerss and Jardine, “Simplicial Homotopy Theory”, V.3, V.4, and V.5. | |
| Feb 21, 2019 at 11:15 | comment | added | Praphulla Koushik | Can you give a reference or you think it is straight forward. I did not yet try to prove... @DmitriPavlov | |
| S Feb 21, 2019 at 7:21 | history | suggested | Praphulla Koushik | CC BY-SA 4.0 |
added latex to make it easily readable
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| Feb 21, 2019 at 6:07 | review | Suggested edits | |||
| S Feb 21, 2019 at 7:21 | |||||
| Feb 21, 2019 at 5:42 | comment | added | David Roberts♦ | Yes, that's what his answer says. | |
| Feb 21, 2019 at 5:41 | comment | added | Praphulla Koushik | @DmitriPavlov So, you are saying there is a weak equivalence of Lie groupoids $(G\rightrightarrows G)$ and the Lie groupoid representing the stack $\underline{*}\times_{BG}\underline{*}$... Is this the case? | |
| Feb 21, 2019 at 5:38 | comment | added | Praphulla Koushik | @DavidRoberts I saw nlab page... It says, loop space object of $BG$ is the stack $\underline{*}\times_{BG}\underline{*}$... | |
| Feb 20, 2019 at 23:07 | comment | added | Dmitri Pavlov | @PraphullaKoushik: What David Roberts said. In addition, be aware that BG in the hyperlinked question is not BG in your question: the former BG is a space (alias ∞-groupoid), whereas your BG is a Lie groupoid, or, alternatively, a stack of ∞-groupoids on the site of smooth manifolds. The former BG is the shape of the latter, see ncatlab.org/nlab/show/shape+modality. | |
| Feb 20, 2019 at 21:25 | comment | added | David Roberts♦ | Where by "homotopy pullback" Dmitri means the pullback of stacks in the appropriate 2-categorical way (or, if you like, the comma object). The loop space here is not the same as the topological loop space (though there is a relation there too), but a higher-categorical analogue (cf ncatlab.org/nlab/show/loop+space+object). "Weak equivalence" means Morita equivalence of the corresponding Lie groupoids, but as I've said elsewhere, I don't think "Morita equivalence" is the best phrase to use. | |
| Feb 20, 2019 at 19:47 | comment | added | Praphulla Koushik | searching in google said that, "loop space" of space $BG$ is a space that is homotopy equivalent to $G$ (math.stackexchange.com/questions/442805/…) Can you please tell me How should I relate with my question? | |
| Feb 20, 2019 at 19:34 | comment | added | Praphulla Koushik | I will say what I understand.. For Lie group $G$, we have obvious map of stacks $\underline{*}\rightarrow BG$ (which is actually an atlas for $BG$). Consider the $2$-fibre product $\underline{*}\times_{BG}\underline{*}$. Here, $*$ denote singleton space and $\underline{*}$ denote the stack associated to singleton manfold. What does it mean to say a weak equivalence $G\rightarrow \underline{*}\times_{BG}\underline{*}$.. I only know weak (Morita ??) equivalence in case of Lie groupoids.. Does it mean weak equivalence as in page 9 of maths.qmul.ac.uk/~noohi/papers/quick.pdf? | |
| Feb 20, 2019 at 19:11 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |