Double groupoid cohomology and extensions

TL;DR

This paper extends classical group theory results to double groupoids, introducing a cohomology theory for classifying extensions and generalizing it to topological double groupoids using sheaf cohomology.

Contribution

It develops a cohomology double complex for double groupoid extensions and generalizes the theory to topological cases with sheaf cohomology methods.

Findings

Extensions classified by first cohomology group

Introduced sheaf cohomology for topological double groupoids

Generalized discrete double groupoid cohomology to topological setting

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.