Trivializing group actions on braided crossed tensor categories and graded braided tensor categories

Abstract

For an abelian group $A$, we study a close connection between braided $A$-crossed tensor categories with a trivialization of the $A$-action and $A$-graded braided tensor categories. Additionally, we prove that the obstruction to the existence of a trivialization of a categorical group action $T$ on a tensor category $\mathcal{C}$ is given by an element $O(T) \in H^2(G, \operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}}))$. In the case that $O(T) = 0$, the set of obstructions forms a torsor over $\operatorname{Hom}(G, \operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}}))$, where $\operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}})$ is the abelian group of tensor natural automorphisms of the identity. The cohomological interpretation of trivializations, together with the homotopical classification of (faithfully graded) braided $A$-crossed tensor categories developed Etingof et al., allows us to provide a method for the construction of faithfully $A$-graded braided tensor categories. We work out two examples. First, we compute the obstruction to the existence of trivializations for the braided $A$-crossed tensor category associated with a pointed semisimple tensor category. In the second example, we compute explicit formulas for the braided $\mathbb{Z}/2\mathbb{Z}$-crossed structures over Tambara–Yamagami fusion categories and, consequently, a conceptual interpretation of the results by Siehler about the classification of braidings over Tambara–Yamagami categories.