论文标题
对功能神经定理及其变化的统一观点
A Unified View on the Functorial Nerve Theorem and its Variations
论文作者
论文摘要
神经定理是代数拓扑的基本结果，它在主题的计算和应用方面起着核心作用。在拓扑数据分析中，通常需要在适当意义上起作用的神经定理，此外，人们通常需要神经定理的封闭盖以及开放式盖子。虽然长期以来证明这种功能神经定理的技术已经可用，但不幸的是，文献中没有对该主题的通用，明确的处理。我们通过证明各种功能神经定理来解决这一问题。首先，我们展示了如何使用基本技术证明通过欧几里得空间中的封闭凸组和子复合物的简单复合物的封面证明神经定理的盖子。然后，我们使用抽象同义理论的标准技术建立了一个更通用的“统一”神经定理，该神经定理涵盖了许多变体。
The nerve theorem is a basic result of algebraic topology that plays a central role in computational and applied aspects of the subject. In topological data analysis, one often needs a nerve theorem that is functorial in an appropriate sense, and furthermore one often needs a nerve theorem for closed covers as well as for open covers. While the techniques for proving such functorial nerve theorems have long been available, there is unfortunately no general-purpose, explicit treatment of this topic in the literature. We address this by proving a variety of functorial nerve theorems. First, we show how one can use elementary techniques to prove nerve theorems for covers by closed convex sets in Euclidean space, and for covers of a simplicial complex by subcomplexes. Then, we establish a more general, "unified" nerve theorem that subsumes many of the variants, using standard techniques from abstract homotopy theory.
