Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology

TL;DR

This paper explores a topos-theoretic framework for modeling variable topological spaces over time, aiming to unify different approaches in persistent homology through lattice theory.

Contribution

It introduces a topos of sheaves over a Heyting algebra derived from persistence diagrams, proposing a unifying algebraic foundation for various persistent homology methods.

Findings

Develops a topos-theoretic model for persistent homology

Establishes a connection between persistence diagrams and Heyting algebras

Proposes a framework for generalized simplicial homology

Abstract

A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally, persistence is a central tool in topological data analysis, which examines the structure of data through topology. The basic techniques have been extended in several different directions, permuting the encoding of topological features by so called barcodes or equivalently persistence diagrams. The set of points of all such diagrams determines a complete Heyting algebra that can explain aspects of the relations between persistent bars through the algebraic properties of its underlying lattice structure. In this paper, we investigate the topos of sheaves over such algebra, as well as discuss its construction and potential for a generalised simplicial homology over it. In particular we are interested in establishing a topos theoretic unifying theory for the various flavours of persistent homology that have emerged so far, providing a global perspective over the algebraic foundations of applied and computational topology.