Since 1976 J.E. Roberts introduced a non-Abelian 1-cohomology of charge-transporters on the Haag-Kaster networks, and as early as 1990 he proved that this cohomology gives a category equivalent to the one of the DHR sectors of the  (Haag dual) net of the observables on the Minkowski d=1+3.In the DHR framework, the covariance of the sectors by the geometric symmetry is introduced through the vacuum representation and morphisms. Quite recently, with G. Ruzzi and E. Vasselli, motivated by theories on a globally hyperbolic spacetime and by sectors with electric charges, as in the analysis by Buchholz and Roberts, we introduced a novel cohomology covariant under the geometric symmetry, for simply connected spacetimes. I will discuss these recent results and some open problems about non-simply connected spacetimes. The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0