In the applied algebraic topology community, the persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. In this paper, we consider a different, more geometric way of generating persistent homology of metric spaces which arises by first embedding a given metric space into a larger space and then considering thickenings of the original space inside this ambient metric space. In the course of doing this, we construct an appropriate category for studying this notion of persistent homology and show that, in a category theoretic sense, the standard persistent homology of the Vietoris-Rips filtration is isomorphic to our geometric persistent homology provided that the ambient metric space satisfies a property called injectivity. As an application of this isomorphism result we are able to precisely characterize the type of intervals that appear in the persistence barcodes of the Vietoris-Rips filtration of any compact metric space and also to give succinct proofs of the characterization of the persistent homology of products and metric gluings of metric spaces. Our results also permit proving several bounds on the length of intervals in the Vietoris-Rips barcode by other metric invariants. Finally, as another application, we connect this geometric persistent homology to the notion of filling radius of manifolds introduced by Gromov \cite{G07} and show some consequences related to (1) the homotopy type of the Vietoris-Rips complexes of spheres which follow from work of M.~Katz and (2) characterization (rigidity) results for spheres in terms of their Vietoris-Rips persistence barcodes which follow from work of F.~Wilhelm.

翻译：在应用代数拓扑学界中，由Vietoris-Rips单纯过滤诱导的持续同调是捕捉度量空间拓扑信息的标准方法。本文考虑一种不同的、更具几何性的度量空间持续同调生成方式：首先将给定度量空间嵌入到更大空间中，然后在此环境度量空间内考虑原始空间的加厚过程。在此过程中，我们构造了适用于研究这种持续同调概念的范畴，并证明：在范畴论意义上，只要环境度量空间满足称为内射性的性质，Vietoris-Rips过滤的标准持续同调就同构于我们的几何持续同调。作为该同构结果的应用，我们能够精确刻画任意紧度量空间Vietoris-Rips滤过持久条形码中出现的区间类型，并给出度量空间乘积与度量黏合的持续同调特征的简洁证明。我们的结果还允许通过其他度量不变量证明Vietoris-Rips条形码中区间长度的多个界。最后，作为另一应用，我们将这种几何持续同调与Gromov引入的流形填充半径概念\cite{G07}建立联系，并展示以下相关推论：（1）由M. Katz工作导出的球面Vietoris-Rips复形的同伦型；（2）由F. Wilhelm工作导出的关于球面Vietoris-Rips持续条形码的特征（刚性）结果。