2020/10/02 by César Galindo, Galindo, César
Mathematics · #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA
paper · pdf · doi:10.48550/arxiv.2010.00847
17 pages
arxiv created 2020/10/02 · arxiv updated 2020/10/05
For an abelian group A , we study a close connection between braided crossed A -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 monoidal category C is given by an element O(T)∈ H2(G,Aut_⊗(IdC)). In the case that O(T)=0, the set of obstructions form a torsor over Hom(G,Aut_⊗(IdC)), where Aut_⊗(IdC) 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 in arXiv:0909.3140, 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 crossed category associated with a pointed semisimple tensor category. In the second example, we compute explicit formulas for the braided ℤ/2-crossed structures over Tambara-Yamagami fusion categories and, consequently, a conceptual interpretation of the results in arXiv:math/0011037 about the classification of braidings over Tambara-Yamagami categories.