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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
Article

Abstract

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.

Keywords

Nonabelian cohomology Higher topos theory Principal bundles 

Notes

Acknowledgments

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.

References

  1. 1.
    Ando, M., Blumberg, A., Gepner, D.: Twists of \(K\)-theory and \(\mathit{TMF}\). In: Superstrings, geometry, topology, and \(C^*\)-algebras, pp. 27–63. Proc. Symp. Pure Math., vol. 81. American Mathematical Society, Providence, RI (2010)Google Scholar
  2. 2.
    Androulidakis, I.: Classification of extensions of principal bundles and transitive Lie groupoids with prescribed kernel and cokernel. J. Math. Phys. 45, 3995 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Blomgren, M., Chacholski, W.: On the classification of fibrations. arXiv:1206.4443
  4. 4.
    Bouwknegt, P., Carey, A., Mathai, V., Murray, M., Stevenson, D.: Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys. 228, 17–49 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Breen, L.: Bitorseurs et cohomologie non abélienne In: Grothendieck Festschrift, pp. 401–476 (1990)Google Scholar
  6. 6.
    Breen, L.: On the classification of 2-gerbes and 2-stacks. Astérisque No. 225 (1994)Google Scholar
  7. 7.
    Breen, L.: Notes on 1- and 2-gerbes. In: Baez, J., May, J.P. (eds.) Towards higher categories. IMA Volume in Mathematics and its Applications, vol. 152, pp. 193–25. Springer, New York (2010)Google Scholar
  8. 8.
    Bullejos, M., Faro, E., García-Muñoz, M.A.: Homotopy colimits and cohomology with local coefficients. Cahiers 44(1), 63–80 (2003)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Giraud, J.: Cohomologie non-abélienne. In: Die Grundlehren der mathematischen Wissenschaften, vol. 179. Springer, Berlin (1971)Google Scholar
  10. 10.
    Husemöller, D.: Fibre bundles, 3rd edn. In: Graduate Texts in Mathematics, vol. 20. Springer, New York (1994)Google Scholar
  11. 11.
    Jardine, J.F., Luo, Z.: Higher order principal bundles. Math. Proc. Cambridge Philos. Soc. 140(2), 221–243 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Johnstone, P.: Topos theory. In: London Mathematical Society Monographs, vol. 10. Academic Press, London (1977)Google Scholar
  13. 13.
    Lurie, J.: Higher topos theory. In: Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton (2009)Google Scholar
  14. 14.
    Lurie, J.: \((\infty ,2)\)-Categories and the Goodwillie calculus. arXiv:0905.0462
  15. 15.
  16. 16.
    May, J.P.: Simplicial Objects in Algebraic Topology. University of Chicago Press, Chicago (1967)Google Scholar
  17. 17.
    May, J.P., Sigurdsson, J.: Parametrized Homotopy Theory. Math Surveys and Monographs, vol. 132. American Mathematical Society, Providence, RI (2006)Google Scholar
  18. 18.
    Mackenzie, K.: On extensions of principal bundles. Ann. Global Anal. Geom. 6(2), 141–163 (1988)Google Scholar
  19. 19.
    Moerdijk, I.: Classifying spaces and classifying topoi. In: Lecture Notes in Mathematics, vol. 1616. Springer, Berlin (1995)Google Scholar
  20. 20.
    Murray, M.K.: Bundle Gerbes J. Lond. Math. Soc. (2) 54(2), 403–416 (1996)Google Scholar
  21. 21.
    Nikolaus, T., Schreiber, U., Stevenson, D.: Principal \(\infty \)-bundles: presentations. J. Homotopy Relat. Struct. (2014). doi: 10.1007/s40062-014-0077-4
  22. 22.
    Nikolaus, T., Waldorf, K.: Four equivalent models for nonabelian gerbes. Pacific J. Math. 264(2), 355–419 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Nikolaus, T., Waldorf, K.: Lifting problems and transgression for non-abelian gerbes. Adv. Math. 242(1), 50–79 (August 2013)Google Scholar
  24. 24.
    Redden, D.C.: Canonical metric connections associated to string structures, PhD thesis. University of Notre Dame (2006)Google Scholar
  25. 25.
    Rezk, C.: Toposes and homotopy toposes. Lectures at UIUC (2005). http://www.math.uiuc.edu/rezk/homotopy-topos-sketch
  26. 26.
    Rosenberg, J.: Continuous-trace algebras from the bundle theoretic point of view. J. Aust. Math. Soc. Ser. A 47(3), 368–381 (1989)zbMATHCrossRefGoogle Scholar
  27. 27.
    Sati, H., Schreiber, U., Stasheff, J.: Twisted differential string- and fivebrane structures. Commun. Math. Phys. 315(1), 169–213 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Schreiber, U.: Differential cohomology in a cohesive \(\infty \)-topos. arXiv:1310.7930
  29. 29.
    Schreiber, U., Schweigert, C., Waldorf, K.: Unoriented WZW models and holonomy of bundle gerbes. Commun. Math. Phys. 274(1), 31–64 (2007)Google Scholar
  30. 30.
    Stasheff, J.: A classification theorem for fiber spaces. Topology 2, 239–246 (1963)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Stevenson, D.: Bundle 2-gerbes. Proc. Lond.Math. Soc. 88(2), 405–435 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Toën, B., Vezzosi, G.: Homotopical algebraic geometry I, topos theory. Adv. Math. 193(2), 257–372 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Waldorf, K.: A loop space formulation for geometric lifting problems. J. Aust. Math. Soc. 90, 129–144 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Wendt, M.: Classifying spaces and fibrations of simplicial sheaves. J. Homotopy Relat. Struct. 6(1), 1–38 (2010)Google Scholar

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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