n-Groupoids and Stacky Groupoids

Abstract

We discuss two generalizations of Lie groupoids. One consists of Lie $n$-groupoids defined as simplicial manifolds with trivial $\ddot{I}{€}_{k\ge n+1}$. 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.

45 pages, include other categories than a previous paper-- arXiv:math/0609420 [math.DG], edited version, typos removed, more details on axioms of stacky groupoids, modified on Cosk