# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1812.05432