论文标题
持续的捆捆同学
Persistent sheaf cohomology
论文作者
论文摘要
我们通过将持久性的概念应用于捆式协调性来扩展计算拓扑中（CO）同源方法的工具箱。由于（模块的）滑轮将拓扑信息与代数信息相结合，因此它们允许沿代数维度和拓扑维度变化。因此，我们介绍了两种不同的捆式共同体（CO）持久模块的结构。其中之一可以被视为简单或奇异共同体复制模块的构建的自然概括。我们讨论了这两种结构如何相互关系，并表明在某些情况下，我们可以将其中一个减少到另一个。此外，我们表明我们可以结合两个结构，以获得具有拓扑和代数维度的二维（CO）持久模块。我们还表明，持久理论的一些经典结果和方法可以推广到滑轮。我们的结果为简单复合物过滤的持续共同学提供了新的观点。
We expand the toolbox of (co)homological methods in computational topology by applying the concept of persistence to sheaf cohomology. Since sheaves (of modules) combine topological information with algebraic information, they allow for variation along an algebraic dimension and along a topological dimension. Consequently, we introduce two different constructions of sheaf cohomology (co)persistence modules. One of them can be viewed as a natural generalization of the construction of simplicial or singular cohomology copersistence modules. We discuss how both constructions relate to each other and show that, in some cases, we can reduce one of them to the other. Moreover, we show that we can combine both constructions to obtain two-dimensional (co)persistence modules with a topological and an algebraic dimension. We also show that some classical results and methods from persistence theory can be generalized to sheaves. Our results open up a new perspective on persistent cohomology of filtrations of simplicial complexes.
