orbifiber bundle and representation of 2-group

orbifiber bundle and representation of 2-group

[Dongning]

Dear Categorists,

In my recent two papers:
http://arxiv.org/abs/1207.4246
http://arxiv.org/abs/1211.3204
We came across with something we call "orbifiber bundle". An orbifiber
bundle is an orbifold analog of fiber bundle whose fiber and base can
be orbifolds. Precise definition is given in Definition2.41 of
arXiv:1207.4246.

A special case is that the base is a manifold while a fiber is an
orbi-vector space, namely vector space with an (effective) finite
group action. For example, let the base be S^2, and fiber be the
complex plain acted by Z/2Z. An explicit construction of this example
using groupoid can be found in this PPT:
http://www.math.wisc.edu/~dwang/Dongnings_Homepage_files/SeidelPPT.pdf
Generalization of the above example is considered in arXiv:1211.3204.

I talked with people who work on orbifolds, and was told this is new.
I wonder if this has been studied by any categorist or stack
specialist already since it seems so nature.

One possible way it occurs is as the following:
If G is a group, there is the well-known relation between G-principal
bundles and functors from representations of G to G-vector bundles.
Now if we replace G with a 2-group and try to make analog of the
relation, then the above orbifiber bundles occur. Is there any work
done along this direction?

And it will be great to know anything else related as well. Thanks in advance!


Best Regards
Dongning Wang

--
PhD candidate
Math Dept of UW-Madison
www.math.wisc.edu/~dwang


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Re: orbifiber bundle and representation of 2-group

[Urs Schreiber]

Dear Dongning Wang,

an orbifold is a special case of a stack on the relevant site, say
that of smooth manifolds. The general theory of principal bundles,
group representations and associated fiber bundles in the context of
higher stacks is discussed in

Thomas Nikolaus, Urs Schreiber, Danny Stevenson,
"Principal infinity-bundles"
arxiv.org/abs/1207.0248
ncatlab.org/schreiber/show/infinity+bundles

The construction that you are after is in section 4.1 there.

See also section 3.6.10, 3.6.11 of

Urs Schreiber
"Differential cohomology in a cohesive infinity-topos"
ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos

All the best,
Urs


On 1/4/13, Dongning <dwang@math.wisc.edu> wrote:
> Dear Categorists,
>
> In my recent two papers:
> http://arxiv.org/abs/1207.4246
> http://arxiv.org/abs/1211.3204
> We came across with something we call "orbifiber bundle". An orbifiber
> bundle is an orbifold analog of fiber bundle whose fiber and base can
> be orbifolds. Precise definition is given in Definition2.41 of
> arXiv:1207.4246.
>
> A special case is that the base is a manifold while a fiber is an
> orbi-vector space, namely vector space with an (effective) finite
> group action. For example, let the base be S^2, and fiber be the
> complex plain acted by Z/2Z. An explicit construction of this example
> using groupoid can be found in this PPT:
> http://www.math.wisc.edu/~dwang/Dongnings_Homepage_files/SeidelPPT.pdf
> Generalization of the above example is considered in arXiv:1211.3204.
>
> I talked with people who work on orbifolds, and was told this is new.
> I wonder if this has been studied by any categorist or stack
> specialist already since it seems so nature.
>
> One possible way it occurs is as the following:
> If G is a group, there is the well-known relation between G-principal
> bundles and functors from representations of G to G-vector bundles.
> Now if we replace G with a 2-group and try to make analog of the
> relation, then the above orbifiber bundles occur. Is there any work
> done along this direction?
>
> And it will be great to know anything else related as well. Thanks in
> advance!
>
>
> Best Regards
> Dongning Wang
>
> --
> PhD candidate
> Math Dept of UW-Madison
> www.math.wisc.edu/~dwang
>


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