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，由Vietoris-Rips单纯形过滤诱导的持续同调是捕获度量空间拓扑信息的标准方法。本文考虑一种更具几何特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更大空间，再在该环境空间内对原始空间进行加厚处理。在此过程中，我们构建了适用于该持续同调概念的范畴，并表明：从范畴论角度而言，当环境度量空间满足称为内射性的性质时，Vietoris-Rips过滤的标准持续同调与我们的几何持续同调同构。利用该同构结果，我们能够精确刻画任意紧致度量空间Vietoris-Rips过滤的持续条码中间隔类型，同时为度量空间积与度量粘合的持续同调刻画提供简洁证明。我们的结果还可通过其他度量不变量证明Vietoris-Rips条码中间隔长度的若干上界。最后，作为另一应用，我们将该几何持续同调与Gromov引入的流形填充半径概念建立联系，并展示相关推论：(1) 基于M. Katz工作的球面Vietoris-Rips复形的同伦型；(2) 基于F. Wilhelm工作的球面Vietoris-Rips持续条码刻画的刚性结果。