2016/08/24 by Jesús Alonso Ochoa Arango, Arango, Jesús Alonso Ochoa, Alejandro Tiraboschi +1
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Homotopy and Cohomology in Algebraic Topology #math.CT #math.KT
paper · pdf · doi:10.48550/arxiv.1608.06712
arxiv created 2016/08/24 · arxiv updated 2016/08/25
We study extensions of double groupoids in the sense of \citeAN2 and show some classical results of group theory extensions in the case of double groupoids. For it, given a double groupoid (B; V,H; P) acting on an abelian group bundle K → P, we introduce a cohomology double complex, in a similar way as was done in \citeAN2 and we show that the extensions of F by 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 \citeD and Tu \citetu, by means of simplicial methods, we introduce a 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.