Lie 2-algebras of vector fields

TL;DR

This paper proves that the category of vector fields on a geometric stack naturally forms a Lie 2-algebra, confirming a conjecture and linking Lie groupoids, Lie algebroids, and stack theory.

Contribution

It establishes a Lie 2-algebra structure on vector fields of geometric stacks using Lie groupoids, confirming Hepworth's conjecture and demonstrating Morita invariance.

Findings

The category of vector fields on a geometric stack has a Lie 2-algebra structure.

The construction is Morita invariant and well-defined on the underlying stack.

The Lie 2-algebra is built from known Lie brackets and sections of the Lie algebroid.

Abstract

We show that the category of vector fields on a geometric stack has the structure of a Lie 2-algebra. This proves a conjecture of R.~Hepworth. The construction uses a Lie groupoid that presents the geometric stack. We show that the category of vector fields on the Lie groupoid is equivalent to the category of vector fields on the stack. The category of vector fields on the Lie groupoid has a Lie 2-algebra structure built from known (ordinary) Lie brackets on multiplicative vector fields of Mackenzie and Xu and the global sections of the Lie algebroid of the Lie groupoid. After giving a precise formulation of Morita invariance of the construction, we verify that the Lie 2-algebra structure defined in this way is well-defined on the underlying stack.