Ephemeral Modules and Scott Sheaves on a Continuous Poset

TL;DR

This paper extends the concept of ephemeral modules to continuous posets using domain theory and establishes an equivalence between a quotient category of persistence modules and sheaves on the Scott topology, also exploring metric properties.

Contribution

It generalizes ephemeral modules to continuous posets and links persistence modules with sheaves on Scott topology, providing new insights into their metric properties.

Findings

Quotient category of persistence modules is equivalent to sheaves on Scott topology.

Generalization of ephemeral modules to continuous posets.

Analysis of metric properties via this categorical equivalence.

Abstract

By utilizing domain theory, we generalize the notion of an ephemeral module to the so-called continuous posets. We investigate the quotient category of persistence modules by the Serre subcategory of ephemeral modules and show that it is equivalent to the category of sheaves on the Scott topology. Furthermore, we study the metric properties of persistence modules via this equivalence.