n-Groupoids and Stacky Groupoids

TL;DR

This paper explores two generalizations of Lie groupoids—Lie n-groupoids and stacky Lie groupoids—and establishes a correspondence between Lie 2-groupoids and stacky Lie groupoids across different categories.

Contribution

It constructs a one-to-one correspondence between Lie 2-groupoids and stacky Lie groupoids, extending the understanding of higher groupoid equivalences in differential and topological contexts.

Findings

Established a correspondence between Lie 2-groupoids and stacky Lie groupoids.

Extended the equivalence framework to both differential and topological categories.

Described equivalences of higher groupoids in various mathematical categories.

Abstract

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.