Timeline for unit element under map of stacks $B\mathcal{G}\rightarrow B\mathcal{H}$
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Dec 5, 2018 at 21:43 | vote | accept | Praphulla Koushik | ||
| Dec 5, 2018 at 11:22 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| Dec 5, 2018 at 10:46 | comment | added | Praphulla Koushik | @BertramArnold Do you have a reference (paper/notes) where this is done in detail.. I think I need this kind of approach to appreciate more of what is going on.. | |
| Dec 5, 2018 at 9:47 | comment | added | Bertram Arnold | Morphisms of Lie groupoids define morphisms of prestacks and then morphisms of stacks by functoriality of stackification. If you unwrap the definitions this morphism of stacks will send trivialized $\mathcal G$-bundles (coming from maps $X\to \mathcal G_0$) to the corresponding trivialized $\mathcal H$-bundles (postcompose with the object map of the functor to get $X\to\mathcal H_0$). In general choose a local trivialization and patch the corresponding trivialized $\mathcal H$-bundles together. You can do this for the "universal" $\mathcal G$-bundle over $B\mathcal G$ to get the bibundle. | |
| Dec 4, 2018 at 20:59 | comment | added | Praphulla Koushik | @BertramArnold One thing that I do not understand is how did you figure out $F(t:\mathcal{G}_1\rightarrow \mathcal{G}_0)$ has to be? I mean what is your definition of map of stacks $B\mathcal{G}\rightarrow B\mathcal{H}$ associated to a morphism of Lie groupoids $\mathcal{G}\rightarrow \mathcal{H}$. My definition (you can see in edit) involves some quotients and notion of bibundle and all which turns out to be $\mathcal{G}_0\times_{\mathcal{H}_0}\mathcal{H}_1$. Can you say how you associate a $B\mathcal{G}\rightarrow B\mathcal{H}$ given a map $\mathcal{G}\rightarrow \mathcal{H}$ .. | |
| Dec 4, 2018 at 18:40 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| Dec 4, 2018 at 18:34 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| Dec 4, 2018 at 17:06 | answer | added | Bertram Arnold | timeline score: 4 | |
| Dec 4, 2018 at 14:53 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| Dec 4, 2018 at 14:47 | comment | added | Praphulla Koushik | @BertramArnold After writing it in detail, I feel this is not suitable for MO. As I have written anyways, do you want to comment on it? Is this is what you had in mind when you say it follows from definition? | |
| Dec 4, 2018 at 14:45 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| Dec 4, 2018 at 13:59 | comment | added | Praphulla Koushik | @BertramArnold Even in Lie groupoid extension it is not that obvious for me... Please wait for some more time so that I can write down what I am planning to write :D | |
| Dec 4, 2018 at 13:48 | comment | added | Bertram Arnold | As you said a general morphism of stacks can't preserve "unit" principal bundles. If both stacks are presented by groupoids and the morphism is presented by a functor it sends the unit bundle of $\mathcal G$ to the pullback of the unit bundle of $\mathcal H$ along the object map of the functor (which should be something like the formula at the end of my previous comment), so in case of a Lie groupoid extension it does preserve unit bundles. All of this should more or less follow from the definitions, in particular how a Lie functor defines a fibered functor between the presented stacks. | |
| Dec 4, 2018 at 13:13 | comment | added | Praphulla Koushik | In that case, object $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ goes to some thing of the form $Q\rightarrow \mathcal{G}_0$ in $B\mathcal{H}$. As $\mathcal{H}_0=\mathcal{G}_0$, object $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ goes to some thing of the form $Q\rightarrow \mathcal{H}_0$ in $B\mathcal{H}$. Now, I can ask if this $Q\rightarrow \mathcal{H}_0$ is same as that of the unit element $\mathcal{H}_1\rightarrow \mathcal{H}_0$ in $B\mathcal{H}$. This is what I meant to ask. I am editing the question so that it makes some sense. @BertramArnold | |
| Dec 4, 2018 at 13:13 | comment | added | Praphulla Koushik | There is something here which does not make sense. As any map of stacks $B\mathcal{G}\rightarrow B\mathcal{H}$ is fiber preserving, it takes an object $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ in $B\mathcal{G}$ to some thing of the form $Q\rightarrow \mathcal{G}_0$ which can not be $t:\mathcal{H}_1\rightarrow \mathcal{H}_0$. I was having in mind the map of stacks coming from a Lie groupid extension (which is simply a morphism of Lie groupoids with same base i.e., $\mathcal{G}_0=\mathcal{H}_0$ with some extra condition on $\phi:\mathcal{G}_1\rightarrow \mathcal{H}_1$). | |
| Dec 4, 2018 at 12:42 | comment | added | Praphulla Koushik | I think you mean to say "[---] If you present the morphism as a functor $F:\mathcal{G}\rightarrow \mathcal{H}$ [---]"... @BertramArnold | |
| Dec 4, 2018 at 12:39 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| Dec 4, 2018 at 11:53 | comment | added | Bertram Arnold | What do you mean by an element of a stack? For me a stack is a sheaf of groupoids, and the principal bundle $t:\mathcal G_1\to\mathcal G_0$ is an object in the groupoid $B\mathcal G(\mathcal G_0)$. A morphism of stacks $B\mathcal G\to\mathcal BH$ sends this to an object of $B\mathcal H(G_0)$, and if you present the morphism as a functor $F_i:\mathcal G_i\to\mathcal H_i$, you can check that this object is the principal $\mathcal H$-bundle $\mathcal H_1\times_{t,\mathcal H_0,F_0}\mathcal G_0$ (the map $F_i$ and functoriality is used to turn this into a principal bundle). | |
| Dec 4, 2018 at 9:25 | history | asked | Praphulla Koushik | CC BY-SA 4.0 |