论文标题
循环框架的小磁盘代数，Grothendieck-Verdier二元性和处理机构组表示
Cyclic framed little disks algebras, Grothendieck-Verdier duality and handlebody group representations
论文作者
论文摘要
我们在任何对称的单型生物中表征了循环代数和框架小的2二氧化烟斗。循环性以连贯的方式适当处理，即达到连贯的同构。当对称性单体生物体被指定为符合有限条件下的线性类别的某些对称的单体生物学生物，我们证明，循环性关联和周期性框架分别是循环的小2盘代数，分别与Pivotal Grothendieck-verdier类别和型号类别相当Boyarchenko-Drinfeld基于Barr的$ \ star $自主类别的概念。我们使用这些结果和Costello的模块化信封结构来获取两个应用程序以供量子拓扑结构：i）我们从某个线性类别的某个对称的单型单体bicategory中提取一个一致的handlebody组表示系统系统，并显示这将概括为Lyubashenko的handlebashenko的手柄组的一部分。 ii）我们通过在圆上评估从模块化函数中提取的类别建立了一个较大的双重性（没有任何对半透明性的假设），从而概括了蒂尔曼和巴卡洛夫 - 基里洛夫的结果。
We characterize cyclic algebras over the associative and the framed little 2-disks operad in any symmetric monoidal bicategory. The cyclicity is appropriately treated in a coherent way, i.e. up to coherent isomorphism. When the symmetric monoidal bicategory is specified to be a certain symmetric monoidal bicategory of linear categories subject to finiteness conditions, we prove that cyclic associative and cyclic framed little 2-disks algebras, respectively, are equivalent to pivotal Grothendieck-Verdier categories and ribbon Grothendieck-Verdier categories, a type of category that was introduced by Boyarchenko-Drinfeld based on Barr's notion of a $\star$-autonomous category. We use these results and Costello's modular envelope construction to obtain two applications to quantum topology: I) We extract a consistent system of handlebody group representations from any ribbon Grothendieck-Verdier category inside a certain symmetric monoidal bicategory of linear categories and show that this generalizes the handlebody part of Lyubashenko's mapping class group representations. II) We establish a Grothendieck-Verdier duality for the category extracted from a modular functor by evaluation on the circle (without any assumption on semisimplicity), thereby generalizing results of Tillmann and Bakalov-Kirillov.
