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The van Est homomorphism for strict Lie 2-groups

Camilo Angulo, Miquel Cueca

TL;DR

The paper develops a differentiation framework for strict Lie 2-groups by constructing a van Est map from the Bott-Shulman-Stasheff double complex to the Weil algebra of the associated Lie 2-algebra. It introduces a simplicial differentiation via a novel simplicial 2-algebroid, and proves that the van Est map is a cochain (and form) map whose cohomology isomorphisms hold under natural connectivity assumptions. The results yield explicit formulas for the van Est map on cochains and forms, and establish an isomorphism in cohomology up to degree $k$ (injective at degree $k+1$), extending classical van Est theory to the 2-group setting. Applications include differentiating 3-shifted symplectic structures and the Segal 2-form on loop groups, with the van Est image matching canonical pairings and structures on related shifted symplectic spaces. This provides a concrete higher-differentiation framework that connects strict Lie 2-groups, their Lie 2-algebras, and shifted symplectic geometry, with potential extensions to non-strict and higher groupoids.

Abstract

We construct a van Est map for strict Lie 2-groups from the Bott-Shulman-Stasheff double complex of the strict Lie 2-group to the Weil algebra of its associated strict Lie 2-algebra. We show that, under appropriate connectedness assumptions, this map induces isomorphisms in cohomology. As an application, we differentiate the Segal 2-form on the loop group.

The van Est homomorphism for strict Lie 2-groups

TL;DR

The paper develops a differentiation framework for strict Lie 2-groups by constructing a van Est map from the Bott-Shulman-Stasheff double complex to the Weil algebra of the associated Lie 2-algebra. It introduces a simplicial differentiation via a novel simplicial 2-algebroid, and proves that the van Est map is a cochain (and form) map whose cohomology isomorphisms hold under natural connectivity assumptions. The results yield explicit formulas for the van Est map on cochains and forms, and establish an isomorphism in cohomology up to degree (injective at degree ), extending classical van Est theory to the 2-group setting. Applications include differentiating 3-shifted symplectic structures and the Segal 2-form on loop groups, with the van Est image matching canonical pairings and structures on related shifted symplectic spaces. This provides a concrete higher-differentiation framework that connects strict Lie 2-groups, their Lie 2-algebras, and shifted symplectic geometry, with potential extensions to non-strict and higher groupoids.

Abstract

We construct a van Est map for strict Lie 2-groups from the Bott-Shulman-Stasheff double complex of the strict Lie 2-group to the Weil algebra of its associated strict Lie 2-algebra. We show that, under appropriate connectedness assumptions, this map induces isomorphisms in cohomology. As an application, we differentiate the Segal 2-form on the loop group.
Paper Structure (20 sections, 21 theorems, 171 equations)

This paper contains 20 sections, 21 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial objects in different categories
  4. The functors $\mathop{\mathrm{Dec}}\nolimits$ and $\overline{W}$
  5. Strict Lie $2$-groups and their strict Lie $2$-algebras
  6. A simplicial differentiation
  7. The simplicial strict Lie $2$-algebroid
  8. The van Est map
  9. The Weil algebra of a Lie $2$-algebroid
  10. The map
  11. The homotopy
  12. The van Est map at the level of functions
  13. The van Est map at the level of forms
  14. The van Est isomorphism
  15. Applications
  16. ...and 5 more sections

Key Result

Proposition 2.8

Let $K_\bullet$ be a simplicial group. The following statements hold:

Theorems & Definitions (54)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4: Lie $n$-algebras as $NQ$-manifolds
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.8
  • proof
  • Proposition 2.10
  • proof
  • ...and 44 more