Transgression to Loop Spaces and its Inverse, III: Gerbes and Thin Fusion Bundles

@article{Waldorf2011TransgressionTL,
  title={Transgression to Loop Spaces and its Inverse, III: Gerbes and Thin Fusion Bundles},
  author={Konrad Waldorf},
  journal={arXiv: Differential Geometry},
  year={2011}
}
  • K. Waldorf
  • Published 2011
  • Mathematics
  • arXiv: Differential Geometry
We show that the category of abelian gerbes over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These principal bundles are equipped with fusion products and are equivariant with respect to thin homotopies between loops. The equivalence is established by a functor called regression, and complements a similar equivalence for bundles and gerbes equipped with connections, derived previously in Part II of this series of papers. The two… Expand
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References

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Transgression to Loop Spaces and its Inverse, II: Gerbes and Fusion Bundles with Connection
We prove that the category of abelian gerbes with connection over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These bundles are equipped withExpand
Transgression to Loop Spaces and its Inverse, I: Diffeological Bundles and Fusion Maps
We prove that isomorphism classes of principal bundles over a diffeological space are in bijection to certain maps on its free loop space, both in a setup with and without connections on the bundles.Expand
A LOOP SPACE FORMULATION FOR GEOMETRIC LIFTING PROBLEMS
  • K. Waldorf
  • Mathematics
  • Journal of the Australian Mathematical Society
  • 2011
Abstract We review and then combine two aspects of the theory of bundle gerbes. The first concerns lifting bundle gerbes and connections on those, developed by Murray and by Gomi. Lifting gerbesExpand
Lifting Problems and Transgression for Non-Abelian Gerbes
We discuss various lifting and reduction problems for bundles and gerbes in the context of a strict Lie 2-group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen'sExpand
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Abstract The holonomy of an unitary line bundle with connection over some base space B is a U(1)-valued function on the loop space LB. In a parallel manner, the holonomy of a gerbe with connection onExpand
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This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Recent developments in mathematicalExpand
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Just as C principal bundles provide a geometric realisation of two-dimensional integral cohomology; gerbes or sheaves of groupoids, provide a geometric realisation of three dimensional integralExpand
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We construct a connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension. The construction is based on certain structures onExpand
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Diffeological spaces are generalizations of smooth manifolds. In this paper, we study the homotopy theory of diffeological spaces. We begin by proving basic properties of the smooth homotopy groupsExpand
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