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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 31st 2015
   • (edited Jul 31st 2015)
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   Author: Urs
   Format: MarkdownItexIn an application I am running into the following situation: given an $\infty$-topos and 1-group objects $G$ and $A$ in it, and given a cocycle $\mathbf{c} \colon \mathbf{B}G \longrightarrow \mathbf{B}^n A$ one is looking for _1-group_ extensions $p \colon \mathbf{B}\hat G\to \mathbf{B}G$ for which $p^\ast \mathbf{c}$ trivializes. If here the constraint that $\hat G$ be a 1-group (be 0-truncated) were omitted, then of course the universal $\mathbf{B}\hat G$ such that $p^\ast \mathbf{c}$ trivcializes is the homotopy fiber of $\mathbf{c}$. With the 1-group condition included, is there still a way to characterized $\hat G$ as a nice universal construction that is guaranteed to exist? This must be something basic.

   In an application I am running into the following situation:

   given an $\infty$-topos and 1-group objects $G$ and $A$ in it, and given a cocycle

   $\mathbf{c} \colon \mathbf{B}G \longrightarrow \mathbf{B}^n A$

   one is looking for 1-group extensions $p \colon \mathbf{B}\hat G\to \mathbf{B}G$ for which $p^\ast \mathbf{c}$ trivializes.

   If here the constraint that $\hat G$ be a 1-group (be 0-truncated) were omitted, then of course the universal $\mathbf{B}\hat G$ such that $p^\ast \mathbf{c}$ trivcializes is the homotopy fiber of $\mathbf{c}$.

   With the 1-group condition included, is there still a way to characterized $\hat G$ as a nice universal construction that is guaranteed to exist? This must be something basic.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJul 31st 2015
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   Author: Mike Shulman
   Format: MarkdownItexWhy do you think that such a 1-group ought to always exist?

   Why do you think that such a 1-group ought to always exist?

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 31st 2015
   • (edited Jul 31st 2015)
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   Author: DavidRoberts
   Format: MarkdownItexDo you have nontrivial examples in mind? It seems the "topology" (or rather shape) of G needs to be uncomplicated, at the very least.

   Do you have nontrivial examples in mind? It seems the “topology” (or rather shape) of G needs to be uncomplicated, at the very least.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 31st 2015
   • (edited Jul 31st 2015)
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   Author: Urs
   Format: MarkdownItex@Mike, I don't think that, I was asking if this is the case. I am wondering quite generally if there is anything useful that one might say here abstractly, as I happen to be looking at an interesting example where this situation appears. @David, yes, the 4-cocycle on the super-translation group in 11d trivializes on the extension by the super group structure on the "exceptional tangent bundle". I mention this in the thread _[from higher to exceptional geometry](http://nforum.ncatlab.org/discussion/6694/from-higher-to-exceptional-geometry/)_. This turns out to be most profound, so I am wondering if there is some nice general abstract story behind it.

   @Mike, I don’t think that, I was asking if this is the case. I am wondering quite generally if there is anything useful that one might say here abstractly, as I happen to be looking at an interesting example where this situation appears.

   @David, yes, the 4-cocycle on the super-translation group in 11d trivializes on the extension by the super group structure on the “exceptional tangent bundle”. I mention this in the thread from higher to exceptional geometry. This turns out to be most profound, so I am wondering if there is some nice general abstract story behind it.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJul 31st 2015
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   Author: Mike Shulman
   Format: MarkdownItexWell, it seems to be asking for a sort of coreflection of the fiber into 1-types, and coreflections into n-types don't exist in general.

   Well, it seems to be asking for a sort of coreflection of the fiber into 1-types, and coreflections into n-types don’t exist in general.

6. • CommentRowNumber6.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 31st 2015
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   Author: DavidRoberts
   Format: MarkdownItexAh, so the group is (apart from the issue of super points) contractible. It may be special to that case, namely a trivialising cover for the higher bundle classified by c just happens to be 1-truncated and deloopable. It may come down to some representation-theoretic fact, perhaps.

   Ah, so the group is (apart from the issue of super points) contractible. It may be special to that case, namely a trivialising cover for the higher bundle classified by c just happens to be 1-truncated and deloopable. It may come down to some representation-theoretic fact, perhaps.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeAug 1st 2015
   • (edited Aug 1st 2015)
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   Author: Urs
   Format: MarkdownItex> a trivialising cover for the higher bundle classified by c just happens to be 1-truncated and deloopable In the example that I am looking at it just so happens, exactly, but is that observation in any way an example of a more general phenomenon? When you follow the references in what I pointed to, you'll see that the first authors found two variants of that 1-group extension, while later authors saw that these sit in a 1-parameter family, each member shares the same underlying bosonic body. I'd like to know things like whether there could be yet more solutions that look completely different. From looking at how the computation works, it however seems intuitively as if in the bosonic part there is no other choice, so maybe it's actually the universal solution to the problem here. This would be most interesting to know.

   a trivialising cover for the higher bundle classified by c just happens to be 1-truncated and deloopable

   In the example that I am looking at it just so happens, exactly, but is that observation in any way an example of a more general phenomenon?

   When you follow the references in what I pointed to, you’ll see that the first authors found two variants of that 1-group extension, while later authors saw that these sit in a 1-parameter family, each member shares the same underlying bosonic body. I’d like to know things like whether there could be yet more solutions that look completely different. From looking at how the computation works, it however seems intuitively as if in the bosonic part there is no other choice, so maybe it’s actually the universal solution to the problem here. This would be most interesting to know.