Journal of Homotopy and Related Structures

, Volume 10, Issue 4, pp 749–801 | Cite as

Principal \(\infty \)-bundles: general theory

  • Thomas Nikolaus
  • Urs Schreiber
  • Danny Stevenson


The theory of principal bundles makes sense in any \(\infty \)-topos, such as the \(\infty \)-topos of topological, of smooth, or of otherwise geometric \(\infty \)-groupoids/\(\infty \)-stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fiber bundles in terms of associated bundles. For suitable choices of structure \(\infty \)-group \(G\) these \(G\)-principal \(\infty \)-bundles reproduce various higher structures that have been considered in the literature and further generalize these to a full geometric model for twisted higher nonabelian sheaf cohomology. We discuss here this general abstract theory of principal \(\infty \)-bundles, observing that it is intimately related to the axioms that characterize \(\infty \)-toposes. A central result is a natural equivalence between principal \(\infty \)-bundles and intrinsic nonabelian cocycles, implying the classification of principal \(\infty \)-bundles by nonabelian sheaf hyper-cohomology. We observe that the theory of geometric fiber \(\infty \)-bundles associated to principal \(\infty \)-bundles subsumes a theory of \(\infty \)-gerbes and of twisted \(\infty \)-bundles, with twists deriving from local coefficient \(\infty \)-bundles, which we define, relate to extensions of principal \(\infty \)-bundles and show to be classified by a corresponding notion of twisted cohomology, identified with the cohomology of a corresponding slice \(\infty \)-topos.


Nonabelian cohomology Higher topos theory Principal bundles 



The writeup of this article and the companion [21] 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; D.S. gratefully acknowledges the support of the Australian Research Council (grant number DP120100106); U.S. acknowledges the support of the Dutch Research Organization NWO (project number 613.000.802). U.S. thanks Domenico Fiorenza for inspiring discussion about twisted cohomology.


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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.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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