G-gerbes, principal 2-group bundles and characteristic classes

Abstract

Let $G$ be a Lie group and $G\to\Aut(G)$ be the canonical group homomorphism induced by the adjoint action of a group on itself. We give an explicit description of a 1-1 correspondence between Morita equivalence classes of, on the one hand, principal 2-group $[G\to\Aut(G)]$-bundles over Lie groupoids and, on the other hand, $G$-extensions of Lie groupoids (i.e.\ between principal $[G\to\Aut(G)]$-bundles over differentiable stacks and $G$-gerbes over differentiable stacks). This approach also allows us to identify $G$-bound gerbes and $[Z(G)\to 1]$-group bundles over differentiable stacks, where $Z(G)$ is the center of $G$. We also introduce universal characteristic classes for 2-group bundles. For groupoid central $G$-extensions, we introduce Dixmier--Douady classes that can be computed from connection-type data generalizing the ones for bundle gerbes. We prove that these classes coincide with universal characteristic classes. As a corollary, we obtain further that Dixmier--Douady classes are integral.

Presentation improved, 38 pages

References in corpus (7)

• Higher-Dimensional Algebra V: 2-Groups
• From double Lie groupoids to local Lie 2-groupoids
• String topology for stacks
• Integration of Exact Courant Algebroids
• Symplectic structures on the integration of exact Courant algebroids
• Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles
• Group actions on stacks and applications to equivariant string topology for stacks

Cited by in corpus (11)

• Four Equivalent Versions of Non-Abelian Gerbes
• Principal actions of stacky Lie groupoids
• On 2-Holonomy
• Atiyah sequence and Gauge transformations of a principal $2$-bundle over a Lie groupoid
• On two notions of a Gerbe over a stack
• On some cocycles which represent the Dixmier-Douady class in simplicial de Rham complexes
• Model structures and quantum cohomology of higher orbifolds
• Lie groupoids and crossed module-valued gerbes over stacks
• The Dixmier-Douady class in the Simplicial de Rham Complex
• A central $U(1)$-extension of a double Lie groupoid
• Crossed extensions and equivalences of topological 2-groupoids