Double groupoid cohomology and extensions

Abstract: We study extensions of double groupoids in the sense of \cite{AN2} and show some classical results of group theory extensions in the case of double groupoids. For it, given a double groupoid $(\mathcal{B}; \mathcal{V},\mathcal{H}; \mathcal{P})$ \emph{acting} on an abelian group bundle $\mathbf{K} \to \mathcal{P}$, we introduce a cohomology double complex, in a similar way as was done in \cite{AN2} and we show that the extensions of $\mathcal{F}$ by $\mathbf{K}$ are classified by the total first cohomology group of the associated total complex. With the aim to extend the above results to the topological setting, following ideas of Deligne \cite{D} and Tu \cite{tu}, by means of simplicial methods, we introduce a \emph{sheaf cohomology} for topological double groupoids, generalizing the double groupoid cohomological in the discrete case, and we carry out in the topological setting the results obtained for discrete double groupoids.