The Smooth Hom-Stack of an Orbifold

Chapter
Part of the MATRIX Book Series book series (MXBS, volume 1)

Abstract

For a compact manifold M and a differentiable stack Open image in new window presented by a Lie groupoid X, we show the Hom-stack Open image in new window is presented by a Fréchet–Lie groupoid Map(M, X) and so is an infinite-dimensional differentiable stack. We further show that if Open image in new window is an orbifold, presented by a proper étale Lie groupoid, then Map(M, X) is proper étale and so presents an infinite-dimensional orbifold.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This research was supported under the Australian Research Council’s Discovery Projects funding scheme (project numbers DP120100106 and DP130102578).

References

  1. 1.
    Baez, J.C., Hoffnung, A.: Convenient categories of smooth spaces. Trans. Am. Math. Soc. 363(11), 5789–5825 (2011). arXiv:0807.1704 MathSciNetCrossRefGoogle Scholar
  2. 2.
    Behrend, K., Xu, P.: Differentiable stacks and gerbes. J. Symplectic Geom. 9(3), 285–341 (2011). arXiv:math/0605694 MathSciNetCrossRefGoogle Scholar
  3. 3.
    Borzellino, J.E., Brunsden, V.: The stratified structure of spaces of smooth orbifold mappings. Commun. Contemp. Math. 15(5), 1350018, 37 (2013). arXiv:0810.1070 MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, W.: On a notion of maps between orbifolds I. Function spaces. Commun. Contemp. Math. 8(5), 569–620 (2006). arXiv:math/0603671 MathSciNetCrossRefGoogle Scholar
  5. 5.
    Frerick, L.: Extension operators for spaces of infinite differentiable Whitney jets. J. Reine Angew. Math. 602, 123–154 (2007).  https://doi.org/10.1515/CRELLE.2007.005 MathSciNetMATHGoogle Scholar
  6. 6.
    Noohi, B.: Mapping stacks of topological stacks. J. Reine Angew. Math. 646, 117–133 (2010). arXiv:0809.2373
  7. 7.
    Roberts, D.M.: Internal categories, anafunctors and localisation. Theory Appl. Categ. 26(29), 788–829 (2012). arXiv:1101.2363
  8. 8.
    Roberts, D.M., Vozzo, R.F.: Smooth loop stacks of differentiable stacks and gerbes (2016). Preprint. arXiv:1602.07973
  9. 9.
    Stacey, A.: Yet more smooth mapping spaces and their smoothly local properties (2013). Preprint. arXiv:1301.5493
  10. 10.
    Weinmann, T.: Orbifolds in the framework of Lie groupoids. Ph.D. thesis, ETH Zürich (2007).  https://doi.org/10.3929/ethz-a-005540169

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

Personalised recommendations