偏序集上的Gerbes与扭C*-动力系统
Gerbes over posets and twisted C*-dynamical systems
摘要：
生成空间$M$的拓扑的基$Δ$当在包含开子集的情况下被排序时成为偏序集（偏序集）。给定一个在群$G$作用下的不动点空间（典型的是C* 代数）的$Δ$上的预层，一般来说找不到一个以它为不动点预层的$G$ -空间的预层。相反，人们得到了$Δ$上的一个gerbe，即一个“扭曲的预层”，其扭曲由一个具有适当的2-群中的系数的上循环编码。我们给出了一个gerbe的holonomy的概念，在一个非交换的上循环的基本群$π_1(M)$。在C*-代数层次上，完整性导致了扭曲C*-动力系统的一般概念，它基于一般的2-群，而不是通常的伴随作用在底层C*-代数上。作为这些概念的应用，我们研究了群G的预层（DR-预层），证明了DR-预层的对偶对象是$Δ$上的群格.还证明了DR-预层的任意截面定义了$π_1(M)$在Cuntz代数上的扭曲作用。
英文摘要：
A base $Δ$ generating the topology of a space $M$ becomes a partially ordered set (poset), when ordered under inclusion of open subsets. Given a precosheaf over $Δ$ of fixed-point spaces (typically C*-algebras) under the action of a group $G$, in general one cannot find a precosheaf of $G$-spaces having it as fixed-point precosheaf. Rather one gets a gerbe over $Δ$, that is, a "twisted precosheaf" whose twisting is encoded by a cocycle with coefficients in a suitable 2-group. We give a notion of holonomy for a gerbe, in terms of a non-abelian cocycle over the fundamental group $π_1(M)$. At the C*-algebraic level, holonomy leads to a general notion of twisted C*-dynamical system, based on a generic 2-group instead of the usual adjoint action on the underlying C*-algebra. As an application of these notions, we study presheaves of group duals (DR-presheaves) and prove that the dual object of a DR-presheaf is a group gerbe over $Δ$. It is also shown that any section of a DR-presheaf defines a twisted action of $π_1(M)$ on a Cuntz algebra.
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