Journal of Homotopy and Related Structures

, Volume 11, Issue 4, pp 923–942 | Cite as

A bigroupoid’s topology (or, Topologising the homotopy bigroupoid of a space)

Article

Abstract

The fundamental bigroupoid of a topological space is one way of capturing its homotopy 2-type. When the space is semilocally 2-connected, one can lift the construction to a bigroupoid internal to the category of topological spaces, as Brown and Danesh-Naruie lifted the fundamental groupoid to a topological groupoid. For locally relatively contractible spaces the resulting topological bigroupoid is locally trivial in a way analogous to the case of the topologised fundamental groupoid. This is the published version of arXiv:1302.7019.

Keywords

Fundamental bigroupoid Homotopy 2-type Topological bigroupoid 

Mathematics Subject Classification

18D05 22A22 55Q05 

Notes

Acknowledgments

Thanks are due to several anonymous referees who helped beat this article into shape over several iterations, and to Tim Porter for both inviting its submission to this volume and his subsequent patience. Thanks also to Ronnie Brown, whose lovely book on groupoids [4] and topology was influential in my thesis work (of which this paper formed a small part) in ways that are not apparent to the casual observer: Happy Birthday Ronnie!

References

  1. 1.
    Bakovic, I.: Bigroupoid 2-torsors. Ph.D. thesis, Ludwig-Maxmillians-Universität München (2007). Available from http://www.irb.hr/korisnici/ibakovic/
  2. 2.
    Bénabou, J.: Introduction to bicategories. In: Proceedings of the midwest category seminar, Springer Lecture Notes, vol. 47. Springer-Verlag (1967)Google Scholar
  3. 3.
    Brazas, J.: Semicoverings: a generalization of covering space theory. Homol. Homot. Appl. 14(1), 33–63 (2012). arXiv:1108.3021
  4. 4.
    Brown, R.: Topology and groupoids. http://groupoids.org.uk/topgpds.html (2006)
  5. 5.
    Brown, R., Danesh-Naruie, G.: The fundamental groupoid as a topological groupoid. Proc. Edinburgh Math. Soc. 2(19):237–244 (1974/75)Google Scholar
  6. 6.
    Ehresmann, C.: Catégories topologiques et catégories différentiables. In: Colloque Géom. Diff. Globale (Bruxelles, 1958), pp. 137–150. Centre Belge Rech. Math., Louvain (1959)Google Scholar
  7. 7.
    Grothendieck, A.: Letters to L. Breen (1975). Dated 17/2/1975, 1975, 17-19/7/1975. Available from http://www.grothendieckcircle.org/
  8. 8.
    Hardie, K.A., Kamps, K., Kieboom, R.: A homotopy 2-groupoid of a Hausdorff space. Appl. Categ. Struct. 8, 209–234 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hardie, K.A., Kamps, K.H., Kieboom, R.W.: A homotopy bigroupoid of a topological space. Appl. Categ. Struct. 9, 311–327 (2001)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Heredia, B.A.: Higher categorical structures in Algebraic Topology: Classifying spaces and homotopy coherence. Ph.D. thesis, Universidad de Granada (2015). http://hdl.handle.net/10481/40300
  11. 11.
    Leinster, T.: Basic bicategories (1998). arXiv:math.CT/9810017
  12. 12.
    Leinster, T.: Higher operads, higher categories, London Math. Soc. Lecture Note Series, vol. 298. Cambridge University Press (2004). arXiv:math.CT/0305049
  13. 13.
    Roberts, D.M.: Fundamental bigroupoids and 2-covering spaces. Ph.D. thesis, University of Adelaide (2009). Available from http://hdl.handle.net/2440/62680
  14. 14.
    Roberts, D.M.: A topological fibrewise fundamental groupoid. Homol. Homot. Appl. 17(2), 37–51 (2015). arXiv:1411.5779 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Roberts, D.M., Schreiber, U.: The inner automorphism 3-group of a strict 2-group. J. Homotopy Rel. Struct. 3(1), 193–245 (2008). arXiv:0708.1741 MathSciNetMATHGoogle Scholar
  16. 16.
    Stevenson, D.: The geometry of bundle gerbes. Ph.D. thesis, Adelaide University, Department of Pure Mathematics (2000). arXiv:math.DG/0004117
  17. 17.
    Trimble, T.: What are ‘fundamental \(n\)-groupoids’? Seminar at DPMMS, Cambridge, 24 August (1999)Google Scholar
  18. 18.
    User ‘Loop Space’ (http://mathoverflow.net/users/45/loop-space): What is the infinite-dimensional-manifold structure on the space of smooth paths mod thin homotopy? MathOverflow (2010) http://mathoverflow.net/q/17843 (version: 2010-03-17)
  19. 19.
    Wada, H.: Local connectivity of mapping spaces. Duke Math. J. 22, 419–425 (1955)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Tbilisi Centre for Mathematical Sciences 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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