[Paper Review] On the comparison of different notions of geometric categories

Joerg Schuermann|ArXiv.org|Jan 29, 2002

Advanced Topology and Set Theory13 references3 citations

TL;DR

This paper introduces the concept of a 'Nash geometric category' to unify and compare three axiomatic frameworks in real geometry: o-minimal structures, analytic geometric categories (van den Dries-Miller), and $ψ$-sets (Shiota). By restricting analytic geometric categories to Nash manifolds, the author establishes a canonical correspondence between these systems, enabling cross-framework transfer of key results such as triangulability, Whitney stratifications, and definable functions as zero-sets of $C^p$-functions.

ABSTRACT

We explain our notion of a Nash geometric category, which allows an easy comparison between the following different axiomatic notions of geometric categories: o-minimal structures on the real field, analytic geometric categories and X-sets (as defined by van den Dries, Miller and Shiota).

Motivation & Objective

• To resolve the lack of a systematic comparison between o-minimal structures, analytic geometric categories, and $ψ$-sets—three distinct axiomatic frameworks in real geometry.
• To address the gap in transferring advanced results (e.g., curve selection, growth dichotomy, generic triviality) across these frameworks due to differing foundational assumptions.
• To introduce a unifying framework—'Nash geometric categories'—that allows direct comparison and transfer of results between the three systems.
• To demonstrate that the Nash geometric category construction preserves key properties (e.g., stability under projections, proper maps) and enables extension of results from one system to others.

Proposed method

• Define a 'Nash geometric category' as a restriction of van den Dries and Miller’s analytic geometric categories to Nash manifolds, ensuring compatibility with semialgebraic and subanalytic structures.
• Establish a one-to-one correspondence between o-minimal structures on $×$ and Nash geometric categories, using the fact that every o-minimal structure on $×$ induces a unique Nash geometric category.
• Prove that the axioms of $ψ$-sets (Shiota) are equivalent to those of Nash geometric categories under the given construction, via local Nash isomorphisms and boundedness conditions.
• Use the equivalence to transfer results: e.g., if a result holds in o-minimal structures (e.g., closed definable sets as zero-sets of $C^p$-functions), it extends to analytic geometric categories and $ψ$-sets.
• Apply the Nash isomorphism technique to reduce global statements to local ones in affine spaces, leveraging the fact that Nash maps preserve semialgebraic and $ψ$-set properties.
• Verify that the axioms of analytic geometric categories (AG1–AG5) and $ψ$-sets (X1–Xiv) are satisfied under the Nash framework, using stability under products, projections, and proper maps.

Experimental results

Research questions

• RQ1 How can o-minimal structures, analytic geometric categories, and $ψ$-sets be systematically compared despite differing foundational axioms?
• RQ2 Can a common framework be constructed that unifies these three geometric categories and enables transfer of results across them?
• RQ3 To what extent do key results—such as triangulability, Whitney stratifications, and definable functions as zero-sets—hold uniformly across these categories?
• RQ4 What role do Nash manifolds play in reconciling the analytic and model-theoretic approaches to geometric categories?
• RQ5 Can the correspondence between o-minimal structures and analytic geometric categories be extended to include $ψ$-sets via a unified Nash-based formalism?

Key findings

• A Nash geometric category is canonically associated to every o-minimal structure on $×$, providing a bridge between model-theoretic and analytic-geometric frameworks.
• The category of $ψ$-sets is equivalent to the category of Nash geometric categories, establishing a unifying foundation for semialgebraic, subanalytic, and o-minimal geometry.
• Results such as the triangulability of definable maps, uniqueness of triangulations, and Thom’s isotopy lemmas hold uniformly in the Nash geometric category framework.
• The closed definable sets in any such category are precisely the zero-sets of definable $C^p$-functions ($1 \leq p < \infty$), generalizing a result from van den Dries and Miller.
• The generic triviality of definable families and the growth dichotomy in polynomially bounded o-minimal structures extend to the Nash geometric category setting.
• The proof of the equivalence relies on local Nash isomorphisms and boundedness, ensuring that properties defined locally on semialgebraic neighborhoods lift globally via the axioms of the category.