2013/07/16 by Primož Škraba, Škraba, Primož, João Pita Costa +1
Computer Science · Mathematics · #06D20 #13P20 #55N35 #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA) #Topological and Geometric Data Analysis #cs.CG #math.AT #math.RA #msc:06D20 #msc:13P20 #msc:55N35
paper · pdf · doi:10.48550/arxiv.1307.4192
20 pages + appendix
openalex publication_date 2013/07/16 · arxiv created 2014/01/31 · arxiv updated 2014/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The intrinsic connection between lattice theory and topology is fairly well established, For instance, the collection of open subsets of a topological subspace always forms a distributive lattice. Persistent homology has been one of the most prominent areas of research in computational topology in the past 20 years. In this paper we will introduce an alternative interpretation of persistence based on the study of the order structure of its correspondent lattice. Its algorithmic construction leads to two operations on homology groups which describe a diagram of spaces as a complete Heyting algebra, which is a generalization of a Boolean algebra. We investigate some of the properties of this lattice, the algorithmic implications of it, and some possible applications.