Metric spaces and homotopy types

2020/12/16 by John F. Jardine, Jardine, J. F.
Decision Sciences · Mathematics · #55U10 #62R40 #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Fuzzy and Soft Set Theory #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2012.09026

openalex publication_date 2020/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By analogy with methods of Spivak, there is a realization functor which takes a persistence diagram Y in simplicial sets to an extended pseudo-metric space (or ep-metric space) Re(Y). The functor Re has a right adjoint, called the singular functor, which takes an ep-metric space Z to a persistence diagram S(Z). We give an explicit description of Re(Y), and show that it depends only on the 1-skeleton sk₁Y of Y. If X is a totally ordered ep-metric space, then there is an isomorphism Re(V_∗(X)) ≅ X, between the realization of the Vietoris-Rips diagram V_∗(X) and the ep-metric space X. The persistence diagrams V_∗(X) and S(X) are sectionwise equivalent for all such X.

Citations

• UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction

Related

• Properties and Stability of Persistence Matching Diagrams
• Curvature Sets Over Persistence Diagrams
• Persistent homotopy theory
• Metric Geometry of Spaces of Persistence Diagrams
• Basic Metric Geometry of the Bottleneck Distance