n-Groupoids and Stacky Groupoids

Details n-Groupoids and Stacky Groupoids n-Groupoids and Stacky Groupoids

2008-01-14

arxiv.org/abs/0801.2057v2

International Mathematics Research Notices (2009) 2009:4087-4141

Mathematics

Differential Geometry

45 pages, include other categories than a previous paper-- arXiv:math/0609420 [math.DG], edited version, typos removed, more d

Scientific paper

10.1093/imrn/rnp080

We discuss two generalizations of Lie groupoids. One consists of Lie $n$-groupoids defined as simplicial manifolds with trivial $\pi_{k\geq n+1}$. The other consists of stacky Lie groupoids $\cG\rra M$ with $\cG$ a differentiable stack. We build a 1-1 correspondence between Lie 2-groupoids and stacky Lie groupoids up to a certain Morita equivalence. We prove this in a general set-up so that the statement is valid in both differential and topological categories. \Equivalences of higher groupoids in various categories are also described.

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