Equivariance In Higher Geometry

Details Equivariance In Higher Geometry Equivariance In Higher Geometry

2010-04-26

arxiv.org/abs/1004.4558v4

Adv. Math. , 226(4): 3367-3408, 2011

Mathematics

Algebraic Topology

42 pages, minor corrections

Scientific paper

We study (pre-)sheaves in bicategories on geometric categories: smooth manifolds, manifolds with a Lie group action and Lie groupoids. We present three main results: we describe equivariant descent, we generalize the plus construction to our setting and show that the plus construction yields a 2-stackification for 2-prestacks. Finally we show that, for a 2-stack, the pullback functor along a Morita-equivalence of Lie groupoids is an equivalence of bicategories. Our results have direct applications to gerbes and 2-vector bundles. For instance, they allow to construct equivariant gerbes from local data and can be used to simplify the description of the local data. We illustrate the usefulness of our results in a systematic discussion of holonomies for unoriented surfaces.

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