Homotopy classification of braided graded categorical groups

For any group GG, a certain cohomology theory of GG-modules is developed. This cohomology arises from the homotopy theory of GG-spaces and it is called the “abelian cohomology of GG-modules”. Then, as the main results of this paper, natural one-to-one correspondences between elements of the 3rd cohomology groups of GG-modules, GG-equivariant pointed simply-connected homotopy 3-types and equivalence classes of braided GG-graded categorical groups are established. The relationship among all these objects with equivariant quadratic functions between GG-modules is also discussed.