On the Classification of Regular Groupoids

We observe that any regular Lie groupoid G over an manifold M fits into an extension $K \to G \to E$ of a foliation groupoid E by a bundle of connected Lie groups K. If $\FF$ is the foliation on M given by the orbits of E and T is a complete transversal to $\FF$, this extension restricts to T, as an extension $K_{T}\to G_{T}\to E_{T}$ of an \'etale groupoid $E_{T}$ by a bundle of connected groups $K_{T}$. We break up the classification into two parts. On the one hand, we classify the latter extensions of \'etale groupoids by (non-abelian) cohomology classes in a new \v{C}ech cohomology of \'{e}tale groupoids. On the other hand, given K and E and an extension $K_{T}\to G_{T}\to E_{T}$ over T, we present a cohomological obstruction to the problem of whether this is the restriction of an extension $K \to G \to E$ over M; if this obstruction vanishes, all extensions $K \to G \to E$ over M which restrict to a given extension over the transversal together form a principal bundle over a ``group'' of bitorsors under K.