Integration of Lie 2-algebras and their morphisms

TL;DR

This paper demonstrates that two different methods of integrating strict Lie 2-algebras into Lie 2-groups are Morita equivalent, and applies this to integrate non-strict morphisms into generalized morphisms.

Contribution

It establishes the Morita equivalence of two integration procedures for strict Lie 2-algebras and extends the integration to non-strict morphisms.

Findings

The two integration methods are Morita equivalent.

Non-strict morphisms can be integrated into generalized morphisms.

Provides a unified understanding of Lie 2-algebra integration.

Abstract

Given a strict Lie 2-algebra, we can integrate it to a strict Lie 2-group by integrating the corresponding Lie algebra crossed module. On the other hand, the integration procedure of Getzler and Henriques will also produce a 2-group. In this paper, we show that these two integration results are Morita equivalent. As an application, we integrate a non-strict morphism between Lie algebra crossed modules to a generalized morphism between their corresponding Lie group crossed modules.