Lie groupoids and crossed module-valued gerbes over stacks

TL;DR

This paper provides a comprehensive differential geometric framework for describing gerbes valued in crossed modules over stacks using Lie groupoids, establishing equivalence with non-Abelian cohomology.

Contribution

It introduces a Lie groupoid-based approach to define and analyze crossed module-valued gerbes over stacks, unifying and generalizing existing notions.

Findings

Gerbes valued in crossed modules can be described using Lie groupoids.

The construction coincides with non-Abelian cohomology definitions.

Provides a differential geometric perspective on gerbes over stacks.

Abstract

We give a precise and general description of gerbes valued in arbitrary crossed module and over an arbitrary differential stack. We do it using only Lie groupoids, hence ordinary differential geometry. We prove the coincidence with the existing notions by comparing our construction with non-Abelian cohomology.