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n-Groupoids and Stacky Groupoids

Chenchang Zhu

Abstract

We discuss two generalizations of Lie groupoids. One consists of Lie $n$-groupoids defined as simplicial manifolds with trivial $π_{k\geq n+1}$. 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.

n-Groupoids and Stacky Groupoids

Abstract

We discuss two generalizations of Lie groupoids. One consists of Lie -groupoids defined as simplicial manifolds with trivial . The other consists of stacky Lie groupoids with 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.

Paper Structure

This paper contains 18 sections, 25 theorems, 94 equations, 2 tables.

Table of Contents

  1. Introduction
  2. $n$-groupoid objects and morphisms in various categories
  3. Hypercovers of $n$-groupoid objects
  4. Pull-back, generalized morphisms and various Morita equivalences
  5. $\mathop{\mathrm{Cosk}}\nolimits^m$, $\mathop{\mathrm{Sk}}\nolimits^m$ and finite data description
  6. The nerve of $X_2 \Rrightarrow X_1 \Rightarrow X_0$
  7. Stacky groupoids in various categories
  8. Good charts
  9. The inverse map
  10. $2$-groupoids and stacky groupoids
  11. From stacky groupoids to $2$-groupoids
  12. The construction of $X_2 \Rrightarrow X_1 \Rightarrow X_0$
  13. Proof that what we construct is a $2$-groupoid
  14. The coherence conditions
  15. Associativity
  16. ...and 3 more sections

Key Result

Theorem 1.4

There is a one-to-one correspondence between SLie (respectively W-) groupoids and Lie $2$-groupoids (respectively Lie $2$-groupoids whose $X_2$ is étale over $\hom(\Lambda[2,j],X)$) modulo $1$-Morita equivalencesMorita equivalences preserving $X_0$ of Lie $2$-groupoids.

Theorems & Definitions (63)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 53 more