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Zig-Zag Modules: Cosheaves and K-Theory

Ryan E. Grady, Anna Schenfisch

TL;DR

This article gives explicit constructible cosheaves for common data-motivated persistence modules, namely, for modules that arise from zig-zag filtrations, and for augmented persistence modules (which encode the data of instantaneous events).

Abstract

Persistence modules have a natural home in the setting of stratified spaces and constructible cosheaves. In this article, we first give explicit constructible cosheaves for common data-motivated persistence modules, namely, for modules that arise from zig-zag filtrations (including monotone filtrations), and for augmented persistence modules (which encode the data of instantaneous events). We then identify an equivalence of categories between a particular notion of zig-zag modules and the combinatorial entrance path category on stratified $\mathbb{R}$. Finally, we compute the algebraic $K$-theory of generalized zig-zag modules and describe connections to both Euler curves and $K_0$ of the monoid of persistence diagrams as described by Bubenik and Elchesen.

Zig-Zag Modules: Cosheaves and K-Theory

TL;DR

This article gives explicit constructible cosheaves for common data-motivated persistence modules, namely, for modules that arise from zig-zag filtrations, and for augmented persistence modules (which encode the data of instantaneous events).

Abstract

Persistence modules have a natural home in the setting of stratified spaces and constructible cosheaves. In this article, we first give explicit constructible cosheaves for common data-motivated persistence modules, namely, for modules that arise from zig-zag filtrations (including monotone filtrations), and for augmented persistence modules (which encode the data of instantaneous events). We then identify an equivalence of categories between a particular notion of zig-zag modules and the combinatorial entrance path category on stratified . Finally, we compute the algebraic -theory of generalized zig-zag modules and describe connections to both Euler curves and of the monoid of persistence diagrams as described by Bubenik and Elchesen.

Paper Structure

This paper contains 31 sections, 28 theorems, 50 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Why K-theory?
  3. Flavors and History of K-theory
  4. K-theory and persistence
  5. What we do
  6. coSheaves from filtrations
  7. Equivalence Theorem
  8. K-theory of Zig-Zag Modules
  9. Euler Curves and Virtual Diagrams
  10. Constructible coSheaves
  11. Stratified/Constructible Basics
  12. Operations on coSheaves
  13. Entrance Paths and Their Representations
  14. Persistence Modules, Persistence Cosheaves, and Filtrations
  15. Persistent Definitions
  16. ...and 16 more sections

Key Result

Lemma 2.1.13

Let $\mathcal{B}$ be a basis for the topology of the space $X$ and let $\mathcal{F}$ be a cosheaf on the poset determined by $\mathcal{B}$. There is a unique (up to isomorphism) extension of $\mathcal{F}$ to a cosheaf on $X$.

Figures (4)

  • Figure 2.1: A stratified circle, $S^1 \to [1]$ as in Example \ref{['ex:s1']}, where $v \mapsto 0$ and $\alpha \mapsto 1$.
  • Figure 3.1: Examples of open intervals occurring in Construction \ref{['cons:pmcosheaf']}.
  • Figure 3.2: An example of the map $C$. The relevant interval for the point, e.g., $m_2$ is $[2,7)$, since the image of each simplex added in that interval under the filter function $f$ is $m_2$. Then $[2,6)$ is mapped to $m_2$ and $[6,7)$ is mapped to $[m_2, m_3)$.
  • Figure 5.1: The result of applying $\delta$ to the cosheaf shown on the top of the figure is the persistence diagram shown on the bottom. In the middle, we have drawn the associated barcode. In the spirit of carlsson2010zigzag, we have shown all bars as closed intervals to emphasize that they do not necessarially arise from a monotone filtration. Note the presence of length-zero barcodes and on-diagonal points, corresponding to indecomposable elements with a single vector space.

Theorems & Definitions (84)

  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Example 2.1.4
  • Definition 2.1.5
  • Definition 2.1.6
  • Definition 2.1.7
  • Definition 2.1.8
  • Definition 2.1.9
  • Remark 2.1.10
  • ...and 74 more