Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Differentiating $L_\infty$ groupoids

Du Li, Leonid Ryvkin, Arne Wessel, Chenchang Zhu

Abstract

Differentiating an Lie $n$-groupoid via the differential-geometric fat point a priori only yielads a presheaf of graded manifolds. In this article we prove that this presheaf is representable by the tangent complex of the Lie $n$-groupoid. As an immediate consequence we obtain that the tangent complex carries the structure of a Lie $n$-algebroid.

Differentiating $L_\infty$ groupoids

Abstract

Differentiating an Lie -groupoid via the differential-geometric fat point a priori only yielads a presheaf of graded manifolds. In this article we prove that this presheaf is representable by the tangent complex of the Lie -groupoid. As an immediate consequence we obtain that the tangent complex carries the structure of a Lie -algebroid.
Paper Structure (15 sections, 24 theorems, 107 equations)

This paper contains 15 sections, 24 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Lie $n$-groupoids and graded manifolds
  3. Simplicial manifolds and their horns
  4. Graded manifolds and their tangent spaces
  5. Iterated tangent bundles for manifolds
  6. Limits for N-manifolds
  7. Compatible connections for $\mathbb Z$-graded vector bundles
  8. Representability of the tangent functor
  9. Description of $\mathcal{T}(X_\bullet)$ as a limit
  10. The central pullback diagram
  11. The case when $k=1$
  12. Embedding $T[1]^kM/T[k]M$ into $(T[1]^{k-1}M)^{[k-1]}$
  13. A monomorphism of $T[1]^k {\wedge^{k}_{\!\!\!\!0}}X$ to $(T[1]^{k}X_{k-1})^{[k]\backslash 0}$
  14. The induction step
  15. Examples

Key Result

Theorem 1.1

(Theorem [thm:ker-p] Let $D_\bullet$ be the simplicial nerve of the pair groupoid of $D$ (cf. Examples ex:fatpoint and ex:pair), and $X_\bullet$ a Lie $n$-groupoid. Then where $p^k_0: X_k \to {\wedge^{n}_{\!\!\!\!0}}(X)$ is the horn projection (see Eq. eq:horn-proj).

Theorems & Definitions (77)

  • Theorem 1.1
  • Corollary 1.2
  • Example 2.1: Pair groupoids in $\mathcal{C}$
  • Example 2.2: Discrete groupoids in $\mathcal{C}$
  • Definition 2.3
  • Remark 2.4
  • Example 2.5: Lie $1$-groupoids
  • Example 2.6: Lie 1-group
  • Definition 2.7
  • Remark 2.8
  • ...and 67 more