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Assignments to sheaves of pseudometric spaces

Michael Robinson

Abstract

An assignment to a sheaf is the choice of a local section from each open set in the sheaf's base space, without regard to how these local sections are related to one another. This article explains that the consistency radius -- which quantifies the agreement between overlapping local sections in the assignment -- is a continuous map. When thresholded, the consistency radius produces the consistency filtration, which is a filtration of open covers. This article shows that the consistency filtration is a functor that transforms the structure of the sheaf and assignment into a nested set of covers in a structure-preserving way. Furthermore, this article shows that consistency filtration is robust to perturbations, establishing its validity for arbitrarily thresholded, noisy data.

Assignments to sheaves of pseudometric spaces

Abstract

An assignment to a sheaf is the choice of a local section from each open set in the sheaf's base space, without regard to how these local sections are related to one another. This article explains that the consistency radius -- which quantifies the agreement between overlapping local sections in the assignment -- is a continuous map. When thresholded, the consistency radius produces the consistency filtration, which is a filtration of open covers. This article shows that the consistency filtration is a functor that transforms the structure of the sheaf and assignment into a nested set of covers in a structure-preserving way. Furthermore, this article shows that consistency filtration is robust to perturbations, establishing its validity for arbitrarily thresholded, noisy data.

Paper Structure

This paper contains 9 sections, 23 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Interpretation and applications
  3. Historical context
  4. Preliminaries
  5. Sheaves paired with assignments
  6. Coarsening filtrations
  7. Consistency radius on subspaces
  8. Consistency filtrations
  9. Sheaves on partial orders

Key Result

Proposition 1

Robinson_sheafcanon For an assignment $a$ to a sheaf $\mathcal{S}$ of pseudometric spaces on $(X,\mathcal{T})$ in which each restriction map of $\mathcal{S}$ is Lipschitz with constant $K$, then for every global section $s$ of $\mathcal{S}$.

Figures (2)

  • Figure 1: The subspace $(X,d)$ of $\mathbb{R}^2$ in Example \ref{['eg:homeo_intransitive']}
  • Figure 2: The consistency filtration for the assignment in Example \ref{['eg:partial_assignment']}; see Example \ref{['eg:cf_example']} for interpretation.

Theorems & Definitions (68)

  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Proposition 1
  • Remark 2
  • ...and 58 more