  We associate to a braided 2-stack ${\cal C}$ a torsor, endowed with a
symmetric cube structure (or $\Sigma$-structure), whose triviality is
equivalent to the existence on ${\cal C}$ of a fully symmetric monoidal
structure. In order to perform the analogous construction for braided 4-stacks,
it is necessary to introduce the notion of a $\Gamma_3$-torsor pair, which is
in the same relation to cubic forms as torsors with $\Sigma$-structure are to
quadratic ones.
