论文标题
对编织的跨张量类别和分级编织张量类别的琐碎小组行动
Trivializing group actions on braided crossed tensor categories and graded braided tensor categories
论文作者
论文摘要
对于Abelian Group $ a $，我们研究了编织的交叉$ a $类别之间的密切联系，并对$ a $ action和$ a $ a $ a的编织张量张量类别进行了琐碎。此外，我们证明了对单体类别$ \ MATHCAL {C} $在H^2（G，\ operatorNornAme {aut} _ \ otimies imimes（\ operatornearneameameameameameage）中的障碍。在$ o（t）= 0 $的情况下，一组障碍物形成了$ \ peripatorName {hom}（g，\ peripatorNAMe {aut} _ \ otimes（\ operatotorname {id} _ {id} _ {\ mathcal {c}}）$， $ \ operatatorName {aut} _ \ otimes（\ operatatorName {id} _ {\ mathcal {c}}}}）$是该身份的Abelian tensor天然自动形态。 琐碎化的同时解释，以及（忠实分级）编织的$ a $跨张量类别的同位分类：0909.3140，我们允许我们提供一种忠实地构建$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $的编织的张贴量的张量张量的方法。我们编写了两个例子。首先，我们计算出与尖头的半密布张量类别相关的编织交叉类别的琐事存在的障碍。在第二个示例中，我们计算编织的$ \ mathbb {Z}/2 $ - 跨tambara-yamagami融合类别的结构，以及对Arxiv中的概念解释的概念解释：Math/0011037关于Tambara-Yamagami类别的Braidings。
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 $\mathcal{C}$ 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 form 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 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 $\mathbb{Z}/2$-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.
