2024/07/11 by Lukas Waas, Waas, Lukas · 2 citations
Computer Science · Mathematics · #55N31 #Algebraic Topology (math.AT) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2407.08666
openalex publication_date 2024/07/11 · openalex created_date 2024/07/14 · openalex updated_date 2026/07/28
When working with (multi-parameter) persistence modules, one usually makes some type of tameness assumption in order to obtain better control over their algebraic behavior. One such notion is Ezra Millers notion of finite encodability, which roughly states that a persistence module can be obtained by pulling back a finite dimensional persistence module over a finite poset. From the perspective of homological algebra, finitely encodable persistence have an inconvenient property: They do not form an abelian category. Here, we prove that if one restricts to such persistence modules which can be constructed in terms of topologically closed and sufficiently constructible (piecewise linear, semi-algebraic, etc.) upsets then abelianity can be restored.