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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