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Strictification of étale stacky Lie groups

Part of: Families, fibrations Categories and geometry

Published online by Cambridge University Press:  30 November 2011

Giorgio Trentinaglia  and
Chenchang Zhu
Giorgio Trentinaglia
Affiliation:
Courant Research Centre ‘Higher Order Structures’, Georg-August-University Göttingen, Bunsenstrasse 3-5, 37073, Göttingen, Germany (email: [email protected])
Chenchang Zhu
Affiliation:
Courant Research Centre ‘Higher Order Structures’, Georg-August-University Göttingen, Bunsenstrasse 3-5, 37073, Göttingen, Germany (email: [email protected])
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Abstract

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We define stacky Lie groups to be group objects in the 2-category of differentiable stacks. We show that every connected and étale stacky Lie group is equivalent to a crossed module of the form (Γ,G) where Γ is the fundamental group of the given stacky Lie group and G is the connected and simply connected Lie group integrating the Lie algebra of the stacky group. Our result is closely related to a strictification result of Baez and Lauda.

Keywords

stacky Lie groupLie 2-groupcrossed modulesstrictificationcategorified group

MSC classification

Secondary: 18F99: None of the above, but in this section 14D23: Stacks and moduli problems
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 3 , May 2012 , pp. 807 - 834
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[BL04]Baez, J. C. and Lauda, A. D., Higher-dimensional algebra. V. 2-groups, Theory Appl. Categ. 12 (2004), 423–491 (electronic).Google Scholar
[BSCS07]Baez, J. C., Stevenson, D., Crans, A. S. and Schreiber, U., From loop groups to 2-groups, Homology, Homotopy Appl. 9 (2007), 101–135.CrossRefGoogle Scholar
[BX06]Behrend, K. and Xu, P., Differentiable stacks and gerbes, arXiv:math.DG/0605694.Google Scholar
[Blo08]Blohmann, C., Stacky Lie groups, Int. Math. Res. Not. IMRN 2008 (2008), Article ID rnn082, 51 pages.CrossRefGoogle Scholar
[BTW08]Blohmann, C., Tang, X. and Weinstein, A., Hopfish structure and modules over irrational rotation algebras, Contemp. Math. 462 (2008), 23–40.CrossRefGoogle Scholar
[BM96]Brylinski, J.-L. and McLaughlin, D. A., The geometry of degree-4 characteristic classes and of line bundles on loop spaces. II, Duke Math. J. 83 (1996), 105–139.CrossRefGoogle Scholar
[CF01]Cattaneo, A. S. and Felder, G., Poisson sigma models and symplectic groupoids, in Quantization of singular symplectic quotients, Progress in Mathematics, vol. 198 (Birkhäuser, Basel, 2001), 61–93.CrossRefGoogle Scholar
[CF03]Crainic, M. and Fernandes, R. L., Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), 575–620.CrossRefGoogle Scholar
[CF04]Crainic, M. and Fernandes, R. L., Integrability of Poisson brackets, J. Differential Geom. 66(1) (2004), 71–137.CrossRefGoogle Scholar
[CM09]Crainic, M. and Mǎrcuţ, I., A normal form theorem around symplectic leaves, arXiv:1009.2090v2 [math.DG].Google Scholar
[Fri82]Friedlander, E. M., Étale homotopy of simplicial schemes, Annals of Mathematics Studies, vol. 104 (Princeton University Press, Princeton, NJ, 1982).Google Scholar
[Get09]Getzler, E., Lie theory for nilpotent L ∞-algebras, Ann. of Math. (2) 170 (2009), 271–301.CrossRefGoogle Scholar
[Hen08]Henriques, A., Integrating L∞-algebras, Compositio Math. 144 (2008), 1017–1045.CrossRefGoogle Scholar
[Mac71]MacLane, S., Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5 (Springer, New York, 1971).Google Scholar
[Met03]Metzler, D., Topological and smooth stacks, arXiv:math.DG/0306176.Google Scholar
[MM03]Moerdijk, I. and Mrčun, J., Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, vol. 91 (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
[Noo05]Noohi, B., Foundations of topological stacks I, arXiv:math.AG/0503247.Google Scholar
[Sch11]Schommer-Pries, C. J., Central extensions of smooth 2-groups and a finite-dimensional string 2-group, Geom. Topol. 15 (2011), 609–676.CrossRefGoogle Scholar
[ST04]Stolz, S. and Teichner, P., What is an elliptic object?, in Topology, geometry and quantum field theory, London Mathematical Society Lecture Note Series, vol. 308 (Cambridge University Press, Cambridge, 2004), 247–343.CrossRefGoogle Scholar
[TWZ07]Tang, X., Weinstein, A. and Zhu, C., Hopfish algebras, Pacific J. Math. 231 (2007), 193–216.CrossRefGoogle Scholar
[Tre10a]Trentinaglia, G., On the role of effective representations of Lie groupoids, Adv. Math. 225 (2010), 826–858.CrossRefGoogle Scholar
[Tre10b]Trentinaglia, G., Tannaka duality for proper Lie groupoids, J. Pure Appl. Algebra 214 (2010), 750–768.CrossRefGoogle Scholar
[Tre11]Trentinaglia, G., Some remarks on the global structure of proper Lie groupoids in low codimensions, Topology Appl. 158 (2011), 708–717.CrossRefGoogle Scholar
[TZ06a]Tseng, H.-H. and Zhu, C., Integrating Lie algebroids via stacks, Compositio Math. 142 (2006), 251–270.CrossRefGoogle Scholar
[TZ06b]Tseng, H.-H. and Zhu, C., Integrating Poisson manifolds via stacks, Trav. Math. 15 (2006), 285–297.Google Scholar
[Zhu07]Zhu, C., Lie II theorem for Lie algebroids via higher groupoids, arXiv:math/0701024v2 [math.DG].Google Scholar
[Zhu09a]Zhu, C., Kan replacement of simplicial manifolds, Lett. Math. Phys. 90 (2009), 383–405.CrossRefGoogle Scholar
[Zhu09b]Zhu, C., n-groupoids and stacky groupoids, Int. Math. Res. Not. IMRN 2009 (2009), 4087–4141.Google Scholar