2021/01/19 by Henry Adams, Johnathan Bush, Joshua Mirth
Computer Science · Mathematics · #Advanced Topology and Set Theory #Combinatorics #Convex metric space #Geometry #Graph #Homotopy #Homotopy and Cohomology in Algebraic Topology #Mathematics #Metric (unit) #Metric space #Product (mathematics) #Pure mathematics #Simplicial complex #Topological and Geometric Data Analysis #Topology (electrical circuits) #Vertex (graph theory) #math.AT #math.CT #math.MG
paper · pdf · doi:10.4204/eptcs.333.18
published as EPTCS 333, 2021, pp. 261-275 · In Proceedings ACT 2020, arXiv:2101.07888
openalex publication_date 2021/01/19 · arxiv created 2021/01/26 · arxiv updated 2021/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Many simplicial complexes arising in practice have an associated metric space structure on the vertex set but not on the complex, e.g. the Vietoris-Rips complex in applied topology. We formalize a remedy by introducing a category of simplicial metric thickenings whose objects have a natural realization as metric spaces. The properties of this category allow us to prove that, for a large class of thickenings including Vietoris-Rips and Cech thickenings, the product of metric thickenings is homotopy equivalent to the metric thickenings of product spaces, and similarly for wedge sums.