Model categories for o-minimal geometry

TL;DR

This paper introduces a new model category framework for o-minimal geometry, connecting definable sets with homotopical methods, and suggests that its cofibrant objects are weak polytopes.

Contribution

It constructs a novel model category for o-minimal structures, bridging definable sets and homotopy theory with a topos-theoretic approach.

Findings

Defines a model category based on o-minimal definable sets

Shows the category resembles topological spaces but with a topos structure

Identifies cofibrant objects as potential weak polytopes

Abstract

We introduce a model category of spaces based on the definable sets of an o-minimal expansion of a real closed field. As a model category, it resembles the category of topological spaces, but its underlying category is a coherent topos. We will show in future work that its cofibrant objects are precisely the "weak polytopes" of Knebusch.