Interval-sphere model structures

TL;DR

This paper introduces a new combinatorial model categorical framework for interval surgery in persistence theory, linking algebraic cell attachments with tameness and compactness in persistence modules.

Contribution

It defines a novel model structure that contextualizes interval surgery as a model-categorical cell attachment, expanding the theoretical foundation of persistence modules.

Findings

New model structure for interval surgery in persistence theory

Cofibrancy linked to tameness and compactness

Interval surgery interpreted as model-categorical cell attachment

Abstract

The bedrock of persistence theory over a single parameter is decomposition of persistence modules into intervals. In [HLM24], the authors leveraged interval decomposition to produce a cell decomposition of the minimal model of a simply connected copersistent space. The key tool was a technique called interval surgery, which involves the gluing of intervals to a persistent CDGA by means of algebraic cell attachments. In this article, we define a compact, combinatorial model categorical structure that contextualizes interval surgery as a genuine model-categorical cell attachment. We show that our new model structure is neither the injective nor the projective one and that cofibrancy is closely linked to the notion of tameness in persistence theory and algebraic notions of compactness.