On the Lie 2-algebra of sections of an LA-groupoid

TL;DR

This paper introduces a new Lie 2-algebra structure on the category of multiplicative sections of an LA-groupoid, demonstrating its invariance and exploring applications to vector fields on stacks and orbifolds.

Contribution

It establishes a strict Lie 2-algebra structure on multiplicative sections of LA-groupoids and applies this to vector fields on differentiable stacks and quotient stacks.

Findings

The category of multiplicative sections has a natural strict Lie 2-algebra structure.

This structure is Morita invariant.

The space of geometric vector fields forms a Lie algebra in various stack cases.

Abstract

In this work we introduce the category of multiplicative sections of an $\la$-groupoid. We prove that this category carries natural strict Lie 2-algebra structures, which are Morita invariant. As applications, we study the algebraic structure underlying multiplicative vector fields on a Lie groupoid and in particular vector fields on differentiable stacks. We also introduce the notion of geometric vector field on the quotient stack of a Lie groupoid, showing that the space of such vector fields is a Lie algebra. We describe the Lie algebra of geometric vector fields in several cases, including classifying stacks, quotient stacks of regular Lie groupoids and in particular orbifolds, and foliation groupoids.