Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Persistent homology of partially ordered spaces

Cameron Calk, Eric Goubault, Philippe Malbos

TL;DR

It is shown that natural homology may be considered a persistence object, and may be calculated as a colimit of uni-dimensional persistent homologies along traces, and may be calculated as a colimit of uni-dimensional persistent homologies along traces.

Abstract

In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was developed in the context of topological data analysis, in which topological properties of point-cloud data sets are extracted while eliminating noise. In both approaches, the evolution homological properties are tracked through a sequence of inclusions of usual topological spaces. Exploiting this similarity, we show that natural homology may be considered a persistence object, and may be calculated as a colimit of uni-dimensional persistent homologies along traces. Finally, we suggest further links and avenues of future work in this direction.

Persistent homology of partially ordered spaces

TL;DR

It is shown that natural homology may be considered a persistence object, and may be calculated as a colimit of uni-dimensional persistent homologies along traces, and may be calculated as a colimit of uni-dimensional persistent homologies along traces.

Abstract

In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was developed in the context of topological data analysis, in which topological properties of point-cloud data sets are extracted while eliminating noise. In both approaches, the evolution homological properties are tracked through a sequence of inclusions of usual topological spaces. Exploiting this similarity, we show that natural homology may be considered a persistence object, and may be calculated as a colimit of uni-dimensional persistent homologies along traces. Finally, we suggest further links and avenues of future work in this direction.
Paper Structure (25 sections, 10 theorems, 37 equations, 1 figure)

This paper contains 25 sections, 10 theorems, 37 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A motivational example
  3. A sketch of persistence
  4. Persistence complexes
  5. Persistent homology
  6. Classification of persistence module and algorithm for persistence
  7. Natural homology of directed spaces
  8. Directed spaces
  9. Trace category and trace diagrams
  10. Natural homology
  11. Bisimulation
  12. Persistent homology of directed spaces
  13. Natural homology as a persistence object
  14. Persistent homology along a trace
  15. Maximal traces and maximal chains
  16. ...and 10 more sections

Key Result

Lemma 1

Suppose $f: \ \mathcal{X} \rightarrow \mathcal{Y}$ is a directed homeomorphismmorphism of directed spaces with inverse $g: \ \mathcal{Y} \rightarrow \mathcal{X}$. Then the functors $\hbox{$\overrightarrow{H}_{n}(\mathcal{X})$}$ and $\hbox{$\overrightarrow{H}_{n}(\mathcal{Y})$}$ are bisimulation equi

Figures (1)

  • Figure 1: Two essentially different concurrent programs with homeomorphic state spaces.

Theorems & Definitions (21)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Theorem 1
  • Proposition 2
  • Proposition 3
  • proof
  • ...and 11 more