Torsorial actions on G-crossed braided tensor categories

We develop a method for generating the complete set of basic data under the torsorial actions of $H_{[\rho ]}^2(G,\mathcal{A})$ and $H^3(G,\text{U}(1))$ on a $G$-crossed braided tensor category ${\mathcal{C}}_G^{\times }$, where $\mathcal{A}$ is the set of invertible simple objects in the braided tensor category $\mathcal{C}$. When $\mathcal{C}$ is a modular tensor category, the $H_{[\rho ]}^2(G,\mathcal{A})$ and $H^3(G,\text{U}(1))$ torsorial action gives a complete generation of possible $G$-crossed extensions, and hence provides a classification. This torsorial classification can be (partially) collapsed by relabeling equivalences that appear when computing the set of $G$-crossed braided extensions of $\mathcal{C}$. The torsor method presented here reduces these redundancies by systematizing relabelings by $\mathcal{A}$-valued $1$-cochains. We also use our methods to compute the composition rule of these torsor functors.