Integrating morphisms of Lie 2-algebras

Abstract

We show how to integrate a weak morphism of Lie algebra crossed-modules to a weak morphism of Lie 2-groups. To do so we develop a theory of butterflies for 2-term L_infty algebras. In particular, we obtain a new description of the bicategory of 2-term L_infty algebras. We use butterflies to give a functorial construction of connected covers of Lie 2-groups. We also discuss the notion of homotopy fiber of a morphism of 2-term L_infty algebras.