Equivariance In Higher Geometry

TL;DR

This paper develops a higher geometric framework for (pre-)sheaves in bicategories on various geometric categories, establishing equivariant descent, plus construction, and Morita-invariance, with applications to gerbes and 2-vector bundles.

Contribution

It generalizes the plus construction to bicategories, proves its 2-stackification property, and shows Morita-equivalence invariance for 2-stacks, advancing higher geometric and categorical understanding.

Findings

Plus construction yields 2-stackification for 2-prestacks.

Pullback along Morita-equivalence is an equivalence of bicategories.

Framework enables systematic construction of equivariant gerbes and simplifies local data descriptions.

Abstract

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.