Extensions in groupoids
As groupoids generalize groups, motivated by group extension we propose a concept of groupoid extension for Lie groupoids, i.e, \[ \sf 1\to A \to G \to K \to 1 \] where $\sf A,G$ and $\sf K$ are Lie groupoids. Similar to the theory of group extension, we show that the existence of extension is obstructed by a groupoid cohomology of $H^3_{\bar \Lambda}({\sf K},Z_{\sf A})$, and the extensions 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.