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1. • CommentRowNumber1.
   • CommentAuthorPraphulla Koushik
   • CommentTimeJul 7th 2018
   • PermaLink
   Author: Praphulla Koushik
   Format: MarkdownItexI am having difficulty understanding definition of Lie groupoid extension as in https://arxiv.org/abs/math/0511696v5 NLab page https://ncatlab.org/nlab/show/centrally+extended+groupoid is also not easy to understand. Can some one tell me what should be a reasonable definition of Lie groupoid extension.

   I am having difficulty understanding definition of Lie groupoid extension as in https://arxiv.org/abs/math/0511696v5

   NLab page https://ncatlab.org/nlab/show/centrally+extended+groupoid is also not easy to understand.

   Can some one tell me what should be a reasonable definition of Lie groupoid extension.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 7th 2018
   • (edited Jul 7th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe abstract idea is extremely simple: Given your Lie groupoid $\mathcal{G}$, regard it as [[smooth stack]] and consider a morphism of smooth stacks of the form $$ \mathcal{G} \longrightarrow \mathbf{B}^2 U(1) $$ to the [[circle 3-group]]. Then the [[homotopy fiber]] of this morphism is a Lie groupoid central extension $\widehat \mathcal{G}$, and, conversely every central extension arises this way, up to equivalence $$ \array{ \widehat{\mathcal{G}} \\ \downarrow \\ \mathcal{G} &\longrightarrow& \mathbf{B}^2 U(1) } $$ That's all there is. The real question now is with which set of generators and relations for [[smooth stacks]] you would like to work. Depending on this choice, this simple idea can take any number of incarnations, and makes for many publications and PhD theses ;-)

   The abstract idea is extremely simple: Given your Lie groupoid $\mathcal{G}$, regard it as smooth stack and consider a morphism of smooth stacks of the form

   $$\mathcal{G} \longrightarrow \mathbf{B}^2 U(1)$$

   to the circle 3-group. Then the homotopy fiber of this morphism is a Lie groupoid central extension $\widehat \mathcal{G}$, and, conversely every central extension arises this way, up to equivalence

   $$\array{ \widehat{\mathcal{G}} \\ \downarrow \\ \mathcal{G} &\longrightarrow& \mathbf{B}^2 U(1) }$$

   That’s all there is. The real question now is with which set of generators and relations for smooth stacks you would like to work. Depending on this choice, this simple idea can take any number of incarnations, and makes for many publications and PhD theses ;-)

3. • CommentRowNumber3.
   • CommentAuthorPraphulla Koushik
   • CommentTimeJul 7th 2018
   • PermaLink
   Author: Praphulla Koushik
   Format: MarkdownItexI understand what does it mean to say see a Lie groupoid as a smooth stack.. it is denoted by $B\mathcal{G}$ in some papers, it is a category whose objects are Principal $\mathcal{G}$ bundles and morphisms are $\mathcal{G}$ invariant morphisms... I believe this is what you mean when you say smooth stack associated to Lie groupoid... it is actually written as $B\mathcal{G}\rightarrow Man$.. I know what is a morphism of stack but I do not know about this particular stack $B^2 U(1)$ you have written... Then you are saying(not defining) homotopy fiber of this morphism of Lie groupoids is a Lie groupoid central xtnsion.. And you are saying any Lie groupoid central extension has to come like this.. Can you say what is a Lie groupoid central extension, I mean the definition... I have to understand only Lie groupoid extensions but I am also open for understanding some special kind of Lie groupoid extensions I.e., Lie groupoid central extensions (I guess Lie groupoid central extensions as you said are special case of Lie groupoid extensions) Thank you and yes I am looking for some problem for my PhD thesis :) :)

   I understand what does it mean to say see a Lie groupoid as a smooth stack.. it is denoted by $B\mathcal{G}$ in some papers, it is a category whose objects are Principal $\mathcal{G}$ bundles and morphisms are $\mathcal{G}$ invariant morphisms… I believe this is what you mean when you say smooth stack associated to Lie groupoid… it is actually written as $B\mathcal{G}\rightarrow Man$.. I know what is a morphism of stack but I do not know about this particular stack $B^2 U(1)$ you have written… Then you are saying(not defining) homotopy fiber of this morphism of Lie groupoids is a Lie groupoid central xtnsion.. And you are saying any Lie groupoid central extension has to come like this..

   Can you say what is a Lie groupoid central extension, I mean the definition… I have to understand only Lie groupoid extensions but I am also open for understanding some special kind of Lie groupoid extensions I.e., Lie groupoid central extensions (I guess Lie groupoid central extensions as you said are special case of Lie groupoid extensions)

   Thank you and yes I am looking for some problem for my PhD thesis :) :)

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 8th 2018
   • (edited Jul 8th 2018)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex>makes for many publications and PhD theses ;-) surely that's not cynical sarcasm?! :-o

   makes for many publications and PhD theses ;-)

   surely that’s not cynical sarcasm?! :-o

5. • CommentRowNumber5.
   • CommentAuthorPraphulla Koushik
   • CommentTimeJul 8th 2018
   • (edited Jul 8th 2018)
   • PermaLink
   Author: Praphulla Koushik
   Format: MarkdownItexhttps://arxiv.org/pdf/math/0511696.pdf in page 5 they define what is a Lie groupoid extension. Let $X_1\rightrightarrows M$ and $X_2\rightrightarrows M$ be two Lie groupoids. A Lie groupoid extension is a morphism of Lie groupoids $X_1\xrightarrow{\phi} Y_1$ inducing identity map on $M$ where $\phi$ is a fibration. I am not very sure what this means. Does it mean $\phi:X_1\rightarrow Y_1$ just as a map of smooth manifolds is a fibration or Is there any other notion of a morphism of Lie groupoids being a fibration?

   https://arxiv.org/pdf/math/0511696.pdf in page 5 they define what is a Lie groupoid extension.

   Let $X_1\rightrightarrows M$ and $X_2\rightrightarrows M$ be two Lie groupoids.

   A Lie groupoid extension is a morphism of Lie groupoids $X_1\xrightarrow{\phi} Y_1$ inducing identity map on $M$ where $\phi$ is a fibration.

   I am not very sure what this means. Does it mean $\phi:X_1\rightarrow Y_1$ just as a map of smooth manifolds is a fibration or Is there any other notion of a morphism of Lie groupoids being a fibration?

6. • CommentRowNumber6.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 9th 2018
   • (edited Jul 9th 2018)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex>Is there any other notion of a morphism of Lie groupoids being a fibration? [[isofibration]], but this may not be what you want. Without looking at the paper, it could just be that $X_1 \to Y_1$ is meant to be a surjective submersion. PS if you select the option "Markdown+Itex", and include write links like this: `<https://example.com>`, then you get <https://example.com>

   Is there any other notion of a morphism of Lie groupoids being a fibration?

   isofibration, but this may not be what you want. Without looking at the paper, it could just be that $X_1 \to Y_1$ is meant to be a surjective submersion.

   PS if you select the option “Markdown+Itex”, and include write links like this: <https://example.com>, then you get https://example.com

7. • CommentRowNumber7.
   • CommentAuthorPraphulla Koushik
   • CommentTimeJul 9th 2018
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   Author: Praphulla Koushik
   Format: MarkdownItexOh, Ok Ok. Thanks. :)

   Oh, Ok Ok. Thanks. :)