Groupoid models for diagrams of groupoid correspondences and their C*-algebras

群对应图描述了一类非常普遍的动力系统，它包含群的自相似性和拓扑图作为特例。自相似群是动力系统的重要不变量，因为它们是纯代数的，并且仍然可以决定Julia集上的所有动力学。图C*-代数是一类重要的C*-代数，因为它们的许多性质可以从基础图中读出。为了用类似的方法处理更多的C*-代数，近年来引入了几个图C*-代数的推广。群对应图可以统一地处理所有这类例子，并提供更多类似于研究的例子，也就是说，它们继承了图C*-代数的许多重要性质。正如一个群作用可以被编码到一个群胚中，这就产生了一个C*-代数，所以一个群胚对应图可以用一个群胚来描述，称为它的群胚模型。然而，这个群胚的现有构造只在一个额外的假设下起作用。对所有图表进行适当的概括是该项目的一个重要目标。或者，我们可以首先将群对应的图平移到C*-代数或纯代数域，然后将该图映射到一个或多个C*-代数，称为协方差代数。这个协方差代数与群胚模型的群胚C*-代数或Steinberg代数有什么关系？这是该提案中的第二个核心问题。众所周知，这些对象在某些假设下是一致的，但并不总是如此。最后，我们将研究Groupoid模型的一些重要性质，这些性质可以用来判断Groupoid模型是否简单，即它的唯一商是显而易见的。

Diagrams of groupoid correspondences describe a very general kind of dynamical systems, which contains self-similarities of groups and topological graphs as special cases. Self-Similar groups are important invariants of dynamical systems because they are purely algebraic and still may determine all of the dynamics on a Julia set. Graph C*-algebras are an important class of C*-algebras because many of their properties may be read off the underlying graphs. In order to handle more C*-algebras in similar ways, several generalisations of graph C*-algebras have been introduced in recent years. Diagrams of groupoid correspondences allow to treat all these classes of examples in a uniform way and also provide more examples that are similarly amenable to study, that is, they inherit many important properties of graph C*-algebras. Just as a group action may be encoded in a groupoid, which then gives rise to a C*-algebra, so a diagram of groupoid correspondences may be described in one groupoid, called its groupoid model. The existing constructions of this groupoid, however, only work under an extra assumption. A suitable generalisation to all diagrams is an important goal of this project. Alternatively, one may first translate a diagram of groupoid correspondences to the realm of C*-algebras or plain algebras and then map the diagram to a single C*-algebra or algebra, called its a covariance algebra. How is this covariance algebra related to the groupoid C*-algebra or Steinberg algebra of the groupoid model? This is the second central question in this proposal. It is known that these objects agree under some assumptions, but not always. Finally, we will investigate some important properties of groupoid models, which allow to decide, for instance, whether the groupoid model is simple, that is, its only quotients are the obvious ones.