Abstract
For a compact manifold M and a differentiable stack presented by a Lie groupoid X, we show the Hom-stack is presented by a Fréchet–Lie groupoid Map(M, X) and so is an infinite-dimensional differentiable stack. We further show that if is an orbifold, presented by a proper étale Lie groupoid, then Map(M, X) is proper étale and so presents an infinite-dimensional orbifold.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Baez, J.C., Hoffnung, A.: Convenient categories of smooth spaces. Trans. Am. Math. Soc. 363(11), 5789–5825 (2011). arXiv:0807.1704
Behrend, K., Xu, P.: Differentiable stacks and gerbes. J. Symplectic Geom. 9(3), 285–341 (2011). arXiv:math/0605694
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
Chen, W.: On a notion of maps between orbifolds I. Function spaces. Commun. Contemp. Math. 8(5), 569–620 (2006). arXiv:math/0603671
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
Noohi, B.: Mapping stacks of topological stacks. J. Reine Angew. Math. 646, 117–133 (2010). arXiv:0809.2373
Roberts, D.M.: Internal categories, anafunctors and localisation. Theory Appl. Categ. 26(29), 788–829 (2012). arXiv:1101.2363
Roberts, D.M., Vozzo, R.F.: Smooth loop stacks of differentiable stacks and gerbes (2016). Preprint. arXiv:1602.07973
Stacey, A.: Yet more smooth mapping spaces and their smoothly local properties (2013). Preprint. arXiv:1301.5493
Weinmann, T.: Orbifolds in the framework of Lie groupoids. Ph.D. thesis, ETH Zürich (2007). https://doi.org/10.3929/ethz-a-005540169
Acknowledgements
This research was supported under the Australian Research Council’s Discovery Projects funding scheme (project numbers DP120100106 and DP130102578).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Roberts, D.M., Vozzo, R.F. (2018). The Smooth Hom-Stack of an Orbifold. In: de Gier, J., Praeger, C., Tao, T. (eds) 2016 MATRIX Annals. MATRIX Book Series, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-72299-3_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-72299-3_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-72298-6
Online ISBN: 978-3-319-72299-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)