Integration of 2-term representations up to homotopy via 2-functors

TL;DR

This paper develops a framework for integrating 2-term representations up to homotopy of Lie algebroids using 2-functors, connecting algebraic and geometric structures through higher groupoids.

Contribution

It introduces a strict 2-functor approach to integrate representations up to homotopy, linking Weinstein groupoids with gauge 2-groupoids for 2-term complexes.

Findings

Holonomy as a strict 2-functor from Weinstein path 2-groupoid

Isomorphism between 1-truncation of transformation 2-groupoid and Weinstein groupoid

New integration schemes for Lie 2-algebras and string algebras

Abstract

Given a representation up to homotopy of a Lie algebroid on a 2-term complex of vector bundles, we define the corresponding holonomy as a strict 2-functor from a Weinstein path 2-groupoid to the gauge 2-groupoid of the underlying 2-term complex. We construct a corresponding transformation 2-groupoid and we prove that the 1-truncation of this 2-groupoid is isomorphic to the Weinstein groupoid of the VB-algebroid associated to a representation up to homotopy. As applications, we describe alternative integration schemes for semi-direct products of Lie 2-algebras and string algebras.