An axiomatic treatment of Persistent Homology
Sergio Tsuyoshi Ura, Marco Contessoto, Alice Kimie Miwa Libardi
TL;DR
This paper establishes an axiomatic foundation for persistent homology applicable across all degrees, ensuring a rigorous mathematical framework for its properties and behaviors.
Contribution
It introduces an axiomatic approach to persistent homology, proving existence and uniqueness of its axioms in both full and reduced forms.
Findings
Axiomatic framework for persistent homology developed
Existence and uniqueness of axioms proven
Applicable to any homology degree
Abstract
We develop an axiomatic framework for persistent homology in any degree. We prove the existence and uniqueness for both a persistent version of the Eilenberg-Steenrod axioms for classical homology and a reduced version of this set of axioms.
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Taxonomy
TopicsCommutative Algebra and Its Applications · Topological and Geometric Data Analysis · Homotopy and Cohomology in Algebraic Topology