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Lie theory for stacky Lie groupoids

**∞-Lie theory** ## Background ### Smooth structure * generalized smooth space * smooth manifold * diffeological space * Frölicher space * smooth topos * Cahiers topos * smooth ∞-groupoid, concrete smooth ∞-groupoid * synthetic differential ∞-groupoid ### Higher groupoids * ∞-groupoid * groupoid * 2-groupoid * strict ∞-groupoid * crossed complex * ∞-group * simplicial group ### Lie theory * Lie theory * Lie integration, Lie differentiation * Lie's three theorems * Lie theory for stacky Lie groupoids ## ∞-Lie groupoids * ∞-Lie groupoid * strict ∞-Lie groupoid * Lie groupoid * differentiable stack * orbifold * ∞-Lie group * Lie group * simple Lie group, semisimple Lie group * Lie 2-group ## ∞-Lie algebroids * ∞-Lie algebroid * Lie algebroid * Lie ∞-algebroid representation * L-∞-algebra * model structure for L-∞ algebras: on dg-Lie algebras, on dg-coalgebras, on simplicial Lie algebras * Lie algebra * semisimple Lie algebra, compact Lie algebra * Lie 2-algebra * strict Lie 2-algebra * differential crossed module * Lie 3-algebra * differential 2-crossed module * dg-Lie algebra, simplicial Lie algebra * super L-∞ algebra * super Lie algebra ## Formal Lie groupoids * formal group, formal groupoid ## Cohomology * Lie algebra cohomology * Chevalley-Eilenberg algebra * Weil algebra * invariant polynomial * Killing form * nonabelian Lie algebra cohomology ## Homotopy * homotopy groups of a Lie groupoid ## Related topics * ∞-Chern-Weil theory ## Examples ### $\infty$-Lie groupoids * Atiyah Lie groupoid * fundamental ∞-groupoid * path groupoid * path n-groupoid * smooth principal ∞-bundle ### $\infty$-Lie groups * orthogonal group * special orthogonal group * spin group * string 2-group * fivebrane 6-group * unitary group * special unitary group * circle Lie n-group * circle group ### $\infty$-Lie algebroids * tangent Lie algebroid * action Lie algebroid * Atiyah Lie algebroid * symplectic Lie n-algebroid * symplectic manifold * Poisson Lie algebroid * Courant Lie algebroid * generalized complex geometry ### $\infty$-Lie algebras * general linear Lie algebra * orthogonal Lie algebra, special orthogonal Lie algebra * endomorphism L-∞ algebra * automorphism ∞-Lie algebra * string Lie 2-algebra * fivebrane Lie 6-algebra * supergravity Lie 3-algebra * supergravity Lie 6-algebra * line Lie n-algebra

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Definition: Stacky Lie groupoid

A stacky Lie groupoid (in contrast to a Lie groupoid) is an internal groupoid in the category of differentiable stacks, such that the space of objects is an ordinary manifold.

Notice that differentiable stacks are equivalent to Lie groupoids modulo Morita equivalence.

Literature

Stacky Lie groupoids and their Lie theory were introduced and studied by Chenchang Zhu and collaborators.

  • Chenchang Zhu, n-Groupoids and Stacky Groupoids, 2008, International Mathematics Research Notices (2009) 2009:4087-4141; (http://arxiv.org/abs/0801.2057 arXiv:0801.2057). DOI: (http://dx.doi.org/10.1093/imrn/rnp080 10.1093/imrn/rnp080).

  • C.Z., Lie nn-groupoids and stacky Lie groupoids (arXiv)

  • C. Z., Lie II theorem for Lie algebroids via stacky Lie groupoids (arXiv)

  • C. Z. Lie II theorem (pdf slides)

  • Henrique Bursztyn, C.Z., Morita equivalence of Poisson manifold via stack groupoids (arXiv)

Further resources

  • nn-Café blog discussion about this is here.
Revised on February 24, 2010 15:13:58 by Urs Schreiber (131.211.235.127)