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.

神经定理是代数拓扑的基本结果，在该学科的计算和应用方面发挥着核心作用。在拓扑数据分析中，人们经常需要一个适当意义上的函子神经定理，而且经常需要一个闭覆盖和开覆盖的神经定理。虽然证明此类函子神经定理的技术早已存在，但不幸的是，文献中没有对该主题的通用、明确的处理。我们通过证明各种函子神经定理来解决这个问题。首先，我们展示如何使用基本技术来证明欧几里德空间中闭凸集覆盖的神经定理，以及子复形覆盖单纯复形的神经定理。然后，我们使用抽象同伦理论的标准技术建立了一个更通用的“统一”神经定理，该定理包含了许多变体。