Differentiable stacks and morita morphism












2












$begingroup$


I heard that if $[X_0/G_0]$ and $[X_1/G_1]$ are differentiable stacks, then any morphism between them is naturally equivalent to



$$(G_0 rightrightarrows X_0) xleftarrow{simeq} (G_2 rightrightarrows X_2) xrightarrow{F} (G_1 rightrightarrows X_1)$$



where the left arrow is morita morphism and the right one is a Lie groupoid morphism.



I can't find a proof of this. Where can I find a detailed proof of this proposition?






differential-geometry reference-request category-theory groupoids topological-stacks







share|cite|improve this question









$endgroup$












  • $begingroup$
    where did you see this? I am also working on stacks/Lie groupoids..
    $endgroup$
    – Praphulla Koushik
    Jan 12 at 18:20










  • $begingroup$
    @PraphullaKoushik theory.fi.infn.it/seminara/Geometry_of_Strings_and_Fieds/… Page 11
    $endgroup$
    – WWK
    Jan 12 at 20:59










  • $begingroup$
    Do you know that any morphism $Bmathcal{G}rightarrow Bmathcal{H}$ is given by a $mathcal{G}-mathcal{H}$ bibundle??? Any $mathcal{G}-mathcal{H}$ bibundle is given by a map that you have mentioned above... is that ok?
    $endgroup$
    – Praphulla Koushik
    Jan 13 at 1:55










  • $begingroup$
    I don't know that. Please write it as an answer below if you know how to prove the above proposition.
    $endgroup$
    – WWK
    Jan 13 at 4:02










  • $begingroup$
    Did you get to see my answe??
    $endgroup$
    – Praphulla Koushik
    Jan 14 at 4:57
















2












$begingroup$


I heard that if $[X_0/G_0]$ and $[X_1/G_1]$ are differentiable stacks, then any morphism between them is naturally equivalent to



$$(G_0 rightrightarrows X_0) xleftarrow{simeq} (G_2 rightrightarrows X_2) xrightarrow{F} (G_1 rightrightarrows X_1)$$



where the left arrow is morita morphism and the right one is a Lie groupoid morphism.



I can't find a proof of this. Where can I find a detailed proof of this proposition?






differential-geometry reference-request category-theory groupoids topological-stacks







share|cite|improve this question









$endgroup$












  • $begingroup$
    where did you see this? I am also working on stacks/Lie groupoids..
    $endgroup$
    – Praphulla Koushik
    Jan 12 at 18:20










  • $begingroup$
    @PraphullaKoushik theory.fi.infn.it/seminara/Geometry_of_Strings_and_Fieds/… Page 11
    $endgroup$
    – WWK
    Jan 12 at 20:59










  • $begingroup$
    Do you know that any morphism $Bmathcal{G}rightarrow Bmathcal{H}$ is given by a $mathcal{G}-mathcal{H}$ bibundle??? Any $mathcal{G}-mathcal{H}$ bibundle is given by a map that you have mentioned above... is that ok?
    $endgroup$
    – Praphulla Koushik
    Jan 13 at 1:55










  • $begingroup$
    I don't know that. Please write it as an answer below if you know how to prove the above proposition.
    $endgroup$
    – WWK
    Jan 13 at 4:02










  • $begingroup$
    Did you get to see my answe??
    $endgroup$
    – Praphulla Koushik
    Jan 14 at 4:57














2












2








2


1



$begingroup$


I heard that if $[X_0/G_0]$ and $[X_1/G_1]$ are differentiable stacks, then any morphism between them is naturally equivalent to



$$(G_0 rightrightarrows X_0) xleftarrow{simeq} (G_2 rightrightarrows X_2) xrightarrow{F} (G_1 rightrightarrows X_1)$$



where the left arrow is morita morphism and the right one is a Lie groupoid morphism.



I can't find a proof of this. Where can I find a detailed proof of this proposition?






differential-geometry reference-request category-theory groupoids topological-stacks







share|cite|improve this question









$endgroup$




I heard that if $[X_0/G_0]$ and $[X_1/G_1]$ are differentiable stacks, then any morphism between them is naturally equivalent to



$$(G_0 rightrightarrows X_0) xleftarrow{simeq} (G_2 rightrightarrows X_2) xrightarrow{F} (G_1 rightrightarrows X_1)$$



where the left arrow is morita morphism and the right one is a Lie groupoid morphism.



I can't find a proof of this. Where can I find a detailed proof of this proposition?






differential-geometry reference-request category-theory groupoids topological-stacks




differential-geometry reference-request category-theory groupoids topological-stacks






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Feb 21 '16 at 15:38









WWKWWK

1,020722




1,020722












  • $begingroup$
    where did you see this? I am also working on stacks/Lie groupoids..
    $endgroup$
    – Praphulla Koushik
    Jan 12 at 18:20










  • $begingroup$
    @PraphullaKoushik theory.fi.infn.it/seminara/Geometry_of_Strings_and_Fieds/… Page 11
    $endgroup$
    – WWK
    Jan 12 at 20:59










  • $begingroup$
    Do you know that any morphism $Bmathcal{G}rightarrow Bmathcal{H}$ is given by a $mathcal{G}-mathcal{H}$ bibundle??? Any $mathcal{G}-mathcal{H}$ bibundle is given by a map that you have mentioned above... is that ok?
    $endgroup$
    – Praphulla Koushik
    Jan 13 at 1:55










  • $begingroup$
    I don't know that. Please write it as an answer below if you know how to prove the above proposition.
    $endgroup$
    – WWK
    Jan 13 at 4:02










  • $begingroup$
    Did you get to see my answe??
    $endgroup$
    – Praphulla Koushik
    Jan 14 at 4:57


















  • $begingroup$
    where did you see this? I am also working on stacks/Lie groupoids..
    $endgroup$
    – Praphulla Koushik
    Jan 12 at 18:20










  • $begingroup$
    @PraphullaKoushik theory.fi.infn.it/seminara/Geometry_of_Strings_and_Fieds/… Page 11
    $endgroup$
    – WWK
    Jan 12 at 20:59










  • $begingroup$
    Do you know that any morphism $Bmathcal{G}rightarrow Bmathcal{H}$ is given by a $mathcal{G}-mathcal{H}$ bibundle??? Any $mathcal{G}-mathcal{H}$ bibundle is given by a map that you have mentioned above... is that ok?
    $endgroup$
    – Praphulla Koushik
    Jan 13 at 1:55










  • $begingroup$
    I don't know that. Please write it as an answer below if you know how to prove the above proposition.
    $endgroup$
    – WWK
    Jan 13 at 4:02










  • $begingroup$
    Did you get to see my answe??
    $endgroup$
    – Praphulla Koushik
    Jan 14 at 4:57
















$begingroup$
where did you see this? I am also working on stacks/Lie groupoids..
$endgroup$
– Praphulla Koushik
Jan 12 at 18:20




$begingroup$
where did you see this? I am also working on stacks/Lie groupoids..
$endgroup$
– Praphulla Koushik
Jan 12 at 18:20












$begingroup$
@PraphullaKoushik theory.fi.infn.it/seminara/Geometry_of_Strings_and_Fieds/… Page 11
$endgroup$
– WWK
Jan 12 at 20:59




$begingroup$
@PraphullaKoushik theory.fi.infn.it/seminara/Geometry_of_Strings_and_Fieds/… Page 11
$endgroup$
– WWK
Jan 12 at 20:59












$begingroup$
Do you know that any morphism $Bmathcal{G}rightarrow Bmathcal{H}$ is given by a $mathcal{G}-mathcal{H}$ bibundle??? Any $mathcal{G}-mathcal{H}$ bibundle is given by a map that you have mentioned above... is that ok?
$endgroup$
– Praphulla Koushik
Jan 13 at 1:55




$begingroup$
Do you know that any morphism $Bmathcal{G}rightarrow Bmathcal{H}$ is given by a $mathcal{G}-mathcal{H}$ bibundle??? Any $mathcal{G}-mathcal{H}$ bibundle is given by a map that you have mentioned above... is that ok?
$endgroup$
– Praphulla Koushik
Jan 13 at 1:55












$begingroup$
I don't know that. Please write it as an answer below if you know how to prove the above proposition.
$endgroup$
– WWK
Jan 13 at 4:02




$begingroup$
I don't know that. Please write it as an answer below if you know how to prove the above proposition.
$endgroup$
– WWK
Jan 13 at 4:02












$begingroup$
Did you get to see my answe??
$endgroup$
– Praphulla Koushik
Jan 14 at 4:57




$begingroup$
Did you get to see my answe??
$endgroup$
– Praphulla Koushik
Jan 14 at 4:57










1 Answer
1






active

oldest

votes


















0












$begingroup$

Given a Lie groupoid $mathcal{G}=(mathcal{G}_1rightrightarrows mathcal{G}_0)$ we have what is called $Bmathcal{G}$ the stack associated to $mathcal{G}$ which is collection of principal $mathcal{G}$ bundles. This in your notation is $[mathcal{G}_0/mathcal{G}_1]$.



Let $F:Bmathcal{G}rightarrow Bmathcal{H}$ be a morphism of stacks. This stack $Bmathcal{G}$ has a special object $t:mathcal{G}_1rightarrow mathcal{G}_0$, the target map.



Consider image of $t$ under $F$, you get a principal $mathcal{H}$ bundle $F(t:mathcal{G}_1rightarrow mathcal{G}_0)$ of the form $Prightarrow mathcal{G}_0$ (as $F$ is fibre preserving, base manifold of $t$ and that of $F(t)$ has to be same).



As $t:mathcal{G}_1rightarrow mathcal{G}_0$ has a left action of $mathcal{G}$ and as $F$ is a functor, $F(t):Prightarrow mathcal{G}_0$ would also have a left action.



This gives a $mathcal{G}-mathcal{H}$ bibundle $$(mathcal{G}_1rightrightarrows mathcal{G}_0)leftarrow Prightarrow (mathcal{H}_1rightrightarrows mathcal{H}_0)$$



This bibundle is also called as generalzied morphism and this comes from
$$(mathcal{G}_1rightrightarrows mathcal{G}_0)leftarrow (Prightrightarrows M )rightarrow (mathcal{H}_1rightrightarrows mathcal{H}_0)$$



You can look at Orbifolds as Stacks by Eugene Lerman. Do not think this notes is about orbifolds. The word orbifold occurs only once in the note and that too in last but one line of the paper.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I was reading some notes about the bibundle. For the relation between HS bibundles and generalised morphisms of Lie groupoids, see the PhD thesis (Theorem 1.73) arxiv.org/pdf/1512.04209.pdf
    $endgroup$
    – WWK
    Jan 14 at 5:50










  • $begingroup$
    I do not understand what you are saying. I understand that there is a correspondence. Is that your thesis?
    $endgroup$
    – Praphulla Koushik
    Jan 14 at 5:58










  • $begingroup$
    Not my thesis. I add this remark for people who read this post in the future.
    $endgroup$
    – WWK
    Jan 14 at 15:05










  • $begingroup$
    @WWK Oh.. That is nice :) You still work on stacks/Lie groupoids? Are you interested in discussing about these?
    $endgroup$
    – Praphulla Koushik
    Jan 14 at 15:28










  • $begingroup$
    Not working on these now...
    $endgroup$
    – WWK
    Jan 14 at 15:55











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1 Answer
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1 Answer
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0












$begingroup$

Given a Lie groupoid $mathcal{G}=(mathcal{G}_1rightrightarrows mathcal{G}_0)$ we have what is called $Bmathcal{G}$ the stack associated to $mathcal{G}$ which is collection of principal $mathcal{G}$ bundles. This in your notation is $[mathcal{G}_0/mathcal{G}_1]$.



Let $F:Bmathcal{G}rightarrow Bmathcal{H}$ be a morphism of stacks. This stack $Bmathcal{G}$ has a special object $t:mathcal{G}_1rightarrow mathcal{G}_0$, the target map.



Consider image of $t$ under $F$, you get a principal $mathcal{H}$ bundle $F(t:mathcal{G}_1rightarrow mathcal{G}_0)$ of the form $Prightarrow mathcal{G}_0$ (as $F$ is fibre preserving, base manifold of $t$ and that of $F(t)$ has to be same).



As $t:mathcal{G}_1rightarrow mathcal{G}_0$ has a left action of $mathcal{G}$ and as $F$ is a functor, $F(t):Prightarrow mathcal{G}_0$ would also have a left action.



This gives a $mathcal{G}-mathcal{H}$ bibundle $$(mathcal{G}_1rightrightarrows mathcal{G}_0)leftarrow Prightarrow (mathcal{H}_1rightrightarrows mathcal{H}_0)$$



This bibundle is also called as generalzied morphism and this comes from
$$(mathcal{G}_1rightrightarrows mathcal{G}_0)leftarrow (Prightrightarrows M )rightarrow (mathcal{H}_1rightrightarrows mathcal{H}_0)$$



You can look at Orbifolds as Stacks by Eugene Lerman. Do not think this notes is about orbifolds. The word orbifold occurs only once in the note and that too in last but one line of the paper.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I was reading some notes about the bibundle. For the relation between HS bibundles and generalised morphisms of Lie groupoids, see the PhD thesis (Theorem 1.73) arxiv.org/pdf/1512.04209.pdf
    $endgroup$
    – WWK
    Jan 14 at 5:50










  • $begingroup$
    I do not understand what you are saying. I understand that there is a correspondence. Is that your thesis?
    $endgroup$
    – Praphulla Koushik
    Jan 14 at 5:58










  • $begingroup$
    Not my thesis. I add this remark for people who read this post in the future.
    $endgroup$
    – WWK
    Jan 14 at 15:05










  • $begingroup$
    @WWK Oh.. That is nice :) You still work on stacks/Lie groupoids? Are you interested in discussing about these?
    $endgroup$
    – Praphulla Koushik
    Jan 14 at 15:28










  • $begingroup$
    Not working on these now...
    $endgroup$
    – WWK
    Jan 14 at 15:55
















0












$begingroup$

Given a Lie groupoid $mathcal{G}=(mathcal{G}_1rightrightarrows mathcal{G}_0)$ we have what is called $Bmathcal{G}$ the stack associated to $mathcal{G}$ which is collection of principal $mathcal{G}$ bundles. This in your notation is $[mathcal{G}_0/mathcal{G}_1]$.



Let $F:Bmathcal{G}rightarrow Bmathcal{H}$ be a morphism of stacks. This stack $Bmathcal{G}$ has a special object $t:mathcal{G}_1rightarrow mathcal{G}_0$, the target map.



Consider image of $t$ under $F$, you get a principal $mathcal{H}$ bundle $F(t:mathcal{G}_1rightarrow mathcal{G}_0)$ of the form $Prightarrow mathcal{G}_0$ (as $F$ is fibre preserving, base manifold of $t$ and that of $F(t)$ has to be same).



As $t:mathcal{G}_1rightarrow mathcal{G}_0$ has a left action of $mathcal{G}$ and as $F$ is a functor, $F(t):Prightarrow mathcal{G}_0$ would also have a left action.



This gives a $mathcal{G}-mathcal{H}$ bibundle $$(mathcal{G}_1rightrightarrows mathcal{G}_0)leftarrow Prightarrow (mathcal{H}_1rightrightarrows mathcal{H}_0)$$



This bibundle is also called as generalzied morphism and this comes from
$$(mathcal{G}_1rightrightarrows mathcal{G}_0)leftarrow (Prightrightarrows M )rightarrow (mathcal{H}_1rightrightarrows mathcal{H}_0)$$



You can look at Orbifolds as Stacks by Eugene Lerman. Do not think this notes is about orbifolds. The word orbifold occurs only once in the note and that too in last but one line of the paper.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I was reading some notes about the bibundle. For the relation between HS bibundles and generalised morphisms of Lie groupoids, see the PhD thesis (Theorem 1.73) arxiv.org/pdf/1512.04209.pdf
    $endgroup$
    – WWK
    Jan 14 at 5:50










  • $begingroup$
    I do not understand what you are saying. I understand that there is a correspondence. Is that your thesis?
    $endgroup$
    – Praphulla Koushik
    Jan 14 at 5:58










  • $begingroup$
    Not my thesis. I add this remark for people who read this post in the future.
    $endgroup$
    – WWK
    Jan 14 at 15:05










  • $begingroup$
    @WWK Oh.. That is nice :) You still work on stacks/Lie groupoids? Are you interested in discussing about these?
    $endgroup$
    – Praphulla Koushik
    Jan 14 at 15:28










  • $begingroup$
    Not working on these now...
    $endgroup$
    – WWK
    Jan 14 at 15:55














0












0








0





$begingroup$

Given a Lie groupoid $mathcal{G}=(mathcal{G}_1rightrightarrows mathcal{G}_0)$ we have what is called $Bmathcal{G}$ the stack associated to $mathcal{G}$ which is collection of principal $mathcal{G}$ bundles. This in your notation is $[mathcal{G}_0/mathcal{G}_1]$.



Let $F:Bmathcal{G}rightarrow Bmathcal{H}$ be a morphism of stacks. This stack $Bmathcal{G}$ has a special object $t:mathcal{G}_1rightarrow mathcal{G}_0$, the target map.



Consider image of $t$ under $F$, you get a principal $mathcal{H}$ bundle $F(t:mathcal{G}_1rightarrow mathcal{G}_0)$ of the form $Prightarrow mathcal{G}_0$ (as $F$ is fibre preserving, base manifold of $t$ and that of $F(t)$ has to be same).



As $t:mathcal{G}_1rightarrow mathcal{G}_0$ has a left action of $mathcal{G}$ and as $F$ is a functor, $F(t):Prightarrow mathcal{G}_0$ would also have a left action.



This gives a $mathcal{G}-mathcal{H}$ bibundle $$(mathcal{G}_1rightrightarrows mathcal{G}_0)leftarrow Prightarrow (mathcal{H}_1rightrightarrows mathcal{H}_0)$$



This bibundle is also called as generalzied morphism and this comes from
$$(mathcal{G}_1rightrightarrows mathcal{G}_0)leftarrow (Prightrightarrows M )rightarrow (mathcal{H}_1rightrightarrows mathcal{H}_0)$$



You can look at Orbifolds as Stacks by Eugene Lerman. Do not think this notes is about orbifolds. The word orbifold occurs only once in the note and that too in last but one line of the paper.






share|cite|improve this answer









$endgroup$



Given a Lie groupoid $mathcal{G}=(mathcal{G}_1rightrightarrows mathcal{G}_0)$ we have what is called $Bmathcal{G}$ the stack associated to $mathcal{G}$ which is collection of principal $mathcal{G}$ bundles. This in your notation is $[mathcal{G}_0/mathcal{G}_1]$.



Let $F:Bmathcal{G}rightarrow Bmathcal{H}$ be a morphism of stacks. This stack $Bmathcal{G}$ has a special object $t:mathcal{G}_1rightarrow mathcal{G}_0$, the target map.



Consider image of $t$ under $F$, you get a principal $mathcal{H}$ bundle $F(t:mathcal{G}_1rightarrow mathcal{G}_0)$ of the form $Prightarrow mathcal{G}_0$ (as $F$ is fibre preserving, base manifold of $t$ and that of $F(t)$ has to be same).



As $t:mathcal{G}_1rightarrow mathcal{G}_0$ has a left action of $mathcal{G}$ and as $F$ is a functor, $F(t):Prightarrow mathcal{G}_0$ would also have a left action.



This gives a $mathcal{G}-mathcal{H}$ bibundle $$(mathcal{G}_1rightrightarrows mathcal{G}_0)leftarrow Prightarrow (mathcal{H}_1rightrightarrows mathcal{H}_0)$$



This bibundle is also called as generalzied morphism and this comes from
$$(mathcal{G}_1rightrightarrows mathcal{G}_0)leftarrow (Prightrightarrows M )rightarrow (mathcal{H}_1rightrightarrows mathcal{H}_0)$$



You can look at Orbifolds as Stacks by Eugene Lerman. Do not think this notes is about orbifolds. The word orbifold occurs only once in the note and that too in last but one line of the paper.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 13 at 4:20









Praphulla KoushikPraphulla Koushik

29817




29817












  • $begingroup$
    I was reading some notes about the bibundle. For the relation between HS bibundles and generalised morphisms of Lie groupoids, see the PhD thesis (Theorem 1.73) arxiv.org/pdf/1512.04209.pdf
    $endgroup$
    – WWK
    Jan 14 at 5:50










  • $begingroup$
    I do not understand what you are saying. I understand that there is a correspondence. Is that your thesis?
    $endgroup$
    – Praphulla Koushik
    Jan 14 at 5:58










  • $begingroup$
    Not my thesis. I add this remark for people who read this post in the future.
    $endgroup$
    – WWK
    Jan 14 at 15:05










  • $begingroup$
    @WWK Oh.. That is nice :) You still work on stacks/Lie groupoids? Are you interested in discussing about these?
    $endgroup$
    – Praphulla Koushik
    Jan 14 at 15:28










  • $begingroup$
    Not working on these now...
    $endgroup$
    – WWK
    Jan 14 at 15:55


















  • $begingroup$
    I was reading some notes about the bibundle. For the relation between HS bibundles and generalised morphisms of Lie groupoids, see the PhD thesis (Theorem 1.73) arxiv.org/pdf/1512.04209.pdf
    $endgroup$
    – WWK
    Jan 14 at 5:50










  • $begingroup$
    I do not understand what you are saying. I understand that there is a correspondence. Is that your thesis?
    $endgroup$
    – Praphulla Koushik
    Jan 14 at 5:58










  • $begingroup$
    Not my thesis. I add this remark for people who read this post in the future.
    $endgroup$
    – WWK
    Jan 14 at 15:05










  • $begingroup$
    @WWK Oh.. That is nice :) You still work on stacks/Lie groupoids? Are you interested in discussing about these?
    $endgroup$
    – Praphulla Koushik
    Jan 14 at 15:28










  • $begingroup$
    Not working on these now...
    $endgroup$
    – WWK
    Jan 14 at 15:55
















$begingroup$
I was reading some notes about the bibundle. For the relation between HS bibundles and generalised morphisms of Lie groupoids, see the PhD thesis (Theorem 1.73) arxiv.org/pdf/1512.04209.pdf
$endgroup$
– WWK
Jan 14 at 5:50




$begingroup$
I was reading some notes about the bibundle. For the relation between HS bibundles and generalised morphisms of Lie groupoids, see the PhD thesis (Theorem 1.73) arxiv.org/pdf/1512.04209.pdf
$endgroup$
– WWK
Jan 14 at 5:50












$begingroup$
I do not understand what you are saying. I understand that there is a correspondence. Is that your thesis?
$endgroup$
– Praphulla Koushik
Jan 14 at 5:58




$begingroup$
I do not understand what you are saying. I understand that there is a correspondence. Is that your thesis?
$endgroup$
– Praphulla Koushik
Jan 14 at 5:58












$begingroup$
Not my thesis. I add this remark for people who read this post in the future.
$endgroup$
– WWK
Jan 14 at 15:05




$begingroup$
Not my thesis. I add this remark for people who read this post in the future.
$endgroup$
– WWK
Jan 14 at 15:05












$begingroup$
@WWK Oh.. That is nice :) You still work on stacks/Lie groupoids? Are you interested in discussing about these?
$endgroup$
– Praphulla Koushik
Jan 14 at 15:28




$begingroup$
@WWK Oh.. That is nice :) You still work on stacks/Lie groupoids? Are you interested in discussing about these?
$endgroup$
– Praphulla Koushik
Jan 14 at 15:28












$begingroup$
Not working on these now...
$endgroup$
– WWK
Jan 14 at 15:55




$begingroup$
Not working on these now...
$endgroup$
– WWK
Jan 14 at 15:55


















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