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

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 C is given by an element O(T) is an element of H-2(G, Au-circle times(Id(C))). In the case that O(T) = 0, the set of obstructions forms a torsor over Hom(G, Aut(circle times)(Id(C))), where Aut(circle times)(Id(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 Z/2Z-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.