On fibrations of Lie groupoids

TL;DR

This paper extends the theory of group extensions to Lie groupoids, introducing fibrations called locally topological product Lie groupoid fibrations, and characterizes their existence and classification via groupoid cohomology.

Contribution

It generalizes the concept of group extensions to Lie groupoids, establishing cohomological obstructions and classifications for these fibrations.

Findings

Existence of fibrations is obstructed by a third cohomology group.

Fibrations are classified by a second cohomology group.

Generalizes group extension theory and gerbes over manifolds/groupoids.

Abstract

As groupoids generalize groups, motivated by group extensions we consider a kind of fibrations of Lie groupoids, called locally topological product Lie groupoid fibrations with fiber $\sf A$, i.e., \[ 1\rightarrow {\sf A} \rightarrow {\sf G} \rightarrow {\sf K}\rightarrow 1 \] where $\sf A,\sf G$ and $\sf K$ are Lie groupoids. Similar to the theory of group extensions, we show that the existence of locally topological product Lie groupoid fibrations with fiber $\sf A$ over $\sf K$ is obstructed by a groupoid cohomology of $H^3_{\bar \Lambda}({\sf K},Z_{\sf A})$, and these locally topological product Lie groupoid fibrations are classified by $H^2_{\bar \Lambda}({\sf K},Z_{\sf A})$ once exists. Here $Z_{\sf A}$ is the center of $\sf A$. This generalizes the theory of group extensions, of gerbes over manifolds/groupoids and etc.