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Journal of Homotopy and Related Structures

, Volume 10, Issue 3, pp 565–622 | Cite as

Principal \(\infty \)-bundles: presentations

  • Thomas Nikolaus
  • Urs Schreiber
  • Danny Stevenson
Article

Abstract

We discuss two aspects of the presentation of the theory of principal \(\infty \)-bundles in an \(\infty \)-topos, introduced in Nikolaus et al. (Principal \(\infty \)-bundles: general theory, 2012), in terms of categories of simplicial (pre)sheaves. First we show that over a cohesive site \(C\) and for \(G\) a presheaf of simplicial groups which is \(C\)-acyclic, \(G\)-principal \(\infty \)-bundles over any object in the \(\infty \)-topos over \(C\) are classified by hyper-Čech-cohomology with coefficients in \(G\). Then we show that over a site \(C\) with enough points, principal \(\infty \)-bundles in the \(\infty \)-topos are presented by ordinary simplicial bundles in the sheaf topos that satisfy principality by stalkwise weak equivalences. Finally we discuss explicit details of these presentations for the discrete site (in discrete \(\infty \)-groupoids) and the smooth site (in smooth \(\infty \)-groupoids, generalizing Lie groupoids and differentiable stacks). In the companion article (Nikolaus et al. in Principal \(\infty \)-bundles: examples and applications, 2012) we use these presentations for constructing classes of examples of (twisted) principal \(\infty \)-bundles and for the discussion of various applications.

Keywords

Homotopy theory Topos theory Differential geometry Algebraic topology Principal bundles 

Notes

Acknowledgments

The writeup of this article and the companions [44, 45] was initiated during a visit by the first two authors to the third author’s institution, University of Glasgow, in summer 2011. It was completed in summer 2012 when all three authors were guests at the Erwin Schrödinger Institute in Vienna. The authors gratefully acknowledge the support of the Engineering and Physical Sciences Research Council grant number EP/I010610/1 and the support of the ESI; DS gratefully acknowledges the support of the Australian Research Council (Grant Number DP120100106).

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Copyright information

© Tbilisi Centre for Mathematical Sciences 2014

Authors and Affiliations

  • Thomas Nikolaus
    • 1
  • Urs Schreiber
    • 2
  • Danny Stevenson
    • 3
  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  2. 2.Mathematics InstituteRadboud Universiteit NijmegenNijmegenThe Netherlands
  3. 3.Department of MathematicsUniversity of GlasgowGlasgowUK

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