Braided N-categories and Σ-structures

We associate to any braided 2-groupoid with vanishing intermediate homotopy group a principal bundle (or torsor) endowed with a so-called Σ-structure, and show that this is the natural generalization to the 2-category context of the familiar quadratic invariant describing a braided groupoid. The corresponding structures for higher braided n-groupoids are also examined. This leads to the concept of Γ3-torsor pairs, which are in the same relation to cubic forms as torsors endowed with a Σ-structure are to quadratic ones. The discussion also covers the corresponding properties of braided 2and n-stacks in groupoids.