A differentiable stack is the analogue in differential geometry of an algebraic stack in algebraic geometry. It can be described either as a stack over differentiable manifolds which admits an atlas, or as a Lie groupoid up to Morita equivalence.[1]
Differentiable stacks are particularly useful to handle spaces with singularities (i.e. orbifolds, leaf spaces, quotients), which appear naturally in differential geometry but are not differentiable manifolds. For instance, differentiable stacks have applications in foliation theory,[2] Poisson geometry[3] and twisted K-theory.[4]
^Blohmann, Christian (2008-01-01). "Stacky Lie Groups". International Mathematics Research Notices. 2008. arXiv:math/0702399. doi:10.1093/imrn/rnn082. ISSN 1687-0247.
^Blohmann, Christian; Weinstein, Alan (2008). "Group-like objects in Poisson geometry and algebra". Poisson Geometry in Mathematics and Physics. Contemporary Mathematics. Vol. 450. American Mathematical Society. pp. 25–39. arXiv:math/0701499. doi:10.1090/conm/450. ISBN 978-0-8218-4423-6. S2CID 16778766.
^Tu, Jean-Louis; Xu, Ping; Laurent-Gengoux, Camille (2004-11-01). "Twisted K-theory of differentiable stacks". Annales Scientifiques de l'École Normale Supérieure. 37 (6): 841–910. arXiv:math/0306138. doi:10.1016/j.ansens.2004.10.002. ISSN 0012-9593. S2CID 119606908 – via Numérisation de documents anciens mathématiques. [fr].
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). This definition shows that orbifolds are a particular kind of differentiablestack. Given an orbifold atlas on a space X {\displaystyle X} , one can...
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