The spectrum of the real line

TL;DR

This paper studies the category of linear representations of the real line, classifies injective objects, and explores the topological structure of the spectrum, connecting it to persistence homology and interleaving distance.

Contribution

It classifies indecomposable injective objects in the category of linear representations of a totally ordered set, and links the spectrum to the topology of persistence modules.

Findings

The category is locally coherent.

The spectrum is homeomorphic to an ordered space.

The topology refines the interleaving distance topology.

Abstract

Motivated by the study of persistence modules over the real line, we investigate the category of linear representations of a totally ordered set. We show that this category is locally coherent and we classify the indecomposable injective objects up to isomorphism. These classes form the spectrum, which we show to be homeomorphic to an ordered space. Moreover, as the spectral category turns out to be discrete, the spectrum parametrises all injective objects. Finally, for the case of the real line we show that this topology refines the topology induced by the interleaving distance, which is known from persistence homology.