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For an abelian group $ A $, we study a close connection between braidedcrossed $ A $-categories with a trivialization of the $ A $-action and $ A$-graded braided tensor categories. Additionally, we prove that the obstructionto the existence of a trivialization of a categorical group action $T$ on amonoidal category $\mathcal{C}$ is given by an element $O(T)\inH^2(G,\operatorname{Aut}_\otimes(\operatorname{Id}_{\mathcal{C}}))$. In thecase that $O(T)=0$, the set of obstructions form a torsor over$\operatorname{Hom}(G,\operatorname{Aut}_\otimes(\operatorname{Id}_{\mathcal{C}}))$,where $\operatorname{Aut}_\otimes(\operatorname{Id}_{\mathcal{C}})$ is theabelian group of tensor natural automorphisms of the identity.  The cohomological interpretation of trivializations, together with thehomotopical classification of (faithfully graded) braided $A$-crossed tensorcategories developed in arXiv:0909.3140, allows us to provide a method for theconstruction of faithfully $A$-graded braided tensor categories. We work outtwo examples. First, we compute the obstruction to the existence oftrivializations for the braided crossed category associated with a pointedsemisimple tensor category. In the second example, we compute explicit formulasfor the braided $\mathbb{Z}/2$-crossed structures over Tambara-Yamagami fusioncategories and, consequently, a conceptual interpretation of the results inarXiv:math/0011037 about the classification of braidings over Tambara-Yamagamicategories.

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