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.
Categories: math.CT math.AG