Strictification of etale stacky Lie groups

TL;DR

This paper introduces a strictification process for etale stacky Lie groups, showing they can be represented as crossed modules involving fundamental groups and simply connected Lie groups, linking to existing strictification results.

Contribution

It establishes that connected etale stacky Lie groups are equivalent to crossed modules, providing a new structural understanding and connecting to Baez and Lauda's strictification work.

Findings

Connected etale stacky Lie groups are equivalent to crossed modules.

Every such group can be represented using fundamental groups and simply connected Lie groups.

The result relates to and extends existing strictification theories.

Abstract

We define stacky Lie groups to be group objects in the 2-category of differentiable stacks. We show that every connected and etale stacky Lie group is equivalent to a crossed module of the form (H,G) where H is the fundamental group of the given stacky Lie group and G is the connected and simply connected Lie group integrating the Lie algebra of the stacky group. Our result is closely related to a strictification result of Baez and Lauda.