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.