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

DOI10.2969/JMSJ/85768576arXiv2010.00847WikidataQ114039858 ScholiaQ114039858MaRDI QIDQ6350376

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Publication date: 2 October 2020

Abstract: 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 $mathcalC$ is given by an element $O(T)in{H}^2(G,operatorname{Aut}_{o}times(operatorname{Id}_{mathcalC}))$. In the case that $O(T)=0$, the set of obstructions form a torsor over $operatornameHom(G,operatorname{Aut}_{o}times(operatorname{Id}_{mathcalC}))$, where $operatorname{Aut}_{o}times(operatorname{Id}_{mathcalC})$ 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 $mathbbZ/2$-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.

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