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Lifespan Functors and Natural Dualities in Persistent Homology

Ulrich Bauer, Maximilian Schmahl

TL;DR

Lifetime functors are introduced, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties, and are applied to morphisms in persistent homology that are induced by morphisms between filtrations.

Abstract

We introduce lifespan functors, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties. They can be used to classify injective and projective objects in the category of barcodes and the category of pointwise finite-dimensional persistence modules. They also naturally appear in duality results for absolute and relative versions of persistent (co)homology, generalizing previous results in terms of barcodes. Due to their functoriality, we can apply these results to morphisms in persistent homology that are induced by morphisms between filtrations. This lays the groundwork for the efficient computation of barcodes for images, kernels, and cokernels of such morphisms.

Lifespan Functors and Natural Dualities in Persistent Homology

TL;DR

Lifetime functors are introduced, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties, and are applied to morphisms in persistent homology that are induced by morphisms between filtrations.

Abstract

We introduce lifespan functors, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties. They can be used to classify injective and projective objects in the category of barcodes and the category of pointwise finite-dimensional persistence modules. They also naturally appear in duality results for absolute and relative versions of persistent (co)homology, generalizing previous results in terms of barcodes. Due to their functoriality, we can apply these results to morphisms in persistent homology that are induced by morphisms between filtrations. This lays the groundwork for the efficient computation of barcodes for images, kernels, and cokernels of such morphisms.

Paper Structure

This paper contains 18 sections, 32 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Persistent Homology
  4. Structure of Persistence Modules
  5. Barcodes
  6. Equivalence of Barcodes and Matching Diagrams
  7. Categorical Properties of Matching Diagrams
  8. Dualization for Persistence Modules
  9. Lifespan Functors
  10. Defining Lifespan Functors
  11. Lifespan Functors and Images, Kernels, and Cokernels
  12. Lifespan of Persistence Modules
  13. Lifespan Functors and Barcodes
  14. Lifespan and Dualization
  15. Projectivity, Injectivity, and Lifespan
  16. ...and 3 more sections

Key Result

Theorem 2.5

The functor $\mathcal{F} \colon \mathbf{Mch}^\mathbf{T} \to \mathbf{Vect}^\mathbf{T}$ reflects the property of being isomorphic: If $D$ and $D'$ are matching diagrams with $\mathcal{F}(D)\cong\mathcal{F}(D')$, then already $D\cong D'$.

Figures (1)

  • Figure 1: Lifespan functors applied to an $\mathbb R$-indexed persistence module $V$ with a visualization of their effect on the barcode, according to \ref{['cor:change_in_barcode']}.

Theorems & Definitions (75)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: MR0265428, Theorem 2.7; see also MR0265428, Section 4.8
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 65 more