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All K-theory is squares K-theory

Josefien Kuijper

TL;DR

The paper demonstrates that the K-theory spectra of many geometric assemblers coincide with the K-theory of appropriate squares categories, unifying disparate contexts under a squares K-theory framework. It develops a general two-step comparison via S^ullet_square and T_ullet constructions, yielding equivalences K^square(C) ≃ K(A) for broad classes of assemblers, including polytopes, varieties, and definable sets. A key application lifts the definable Euler characteristic from the level of objects to a map of K-theory spectra, with constructions in o-minimal structures and connections to Z/2-valued constructible functions. The work provides concrete examples and natural derived invariants, such as a derived Euler characteristic, and suggests a flexible, high-level approach to comparing K-theory across geometric settings.

Abstract

We show that the K-theory spectra of many assemblers, such as the assembler of polytopes in euclidean, hyperbolic or spherical geometry, as well as the assembler of definable sets, are equivalent to the K-theory spectrum of a squares category. We use this to lift the definable Euler characteristic of definable sets in an o-minimal structure to a map of K-theory spectra.

All K-theory is squares K-theory

TL;DR

The paper demonstrates that the K-theory spectra of many geometric assemblers coincide with the K-theory of appropriate squares categories, unifying disparate contexts under a squares K-theory framework. It develops a general two-step comparison via S^ullet_square and T_ullet constructions, yielding equivalences K^square(C) ≃ K(A) for broad classes of assemblers, including polytopes, varieties, and definable sets. A key application lifts the definable Euler characteristic from the level of objects to a map of K-theory spectra, with constructions in o-minimal structures and connections to Z/2-valued constructible functions. The work provides concrete examples and natural derived invariants, such as a derived Euler characteristic, and suggests a flexible, high-level approach to comparing K-theory across geometric settings.

Abstract

We show that the K-theory spectra of many assemblers, such as the assembler of polytopes in euclidean, hyperbolic or spherical geometry, as well as the assembler of definable sets, are equivalent to the K-theory spectrum of a squares category. We use this to lift the definable Euler characteristic of definable sets in an o-minimal structure to a map of K-theory spectra.

Paper Structure

This paper contains 13 sections, 27 theorems, 64 equations.

Table of Contents

  1. Motivation
  2. Outline
  3. Application
  4. Acknowledgements
  5. Preliminaries
  6. On the $S_\bullet^\square$-construction for squares categories
  7. Comparing the $S_\bullet^\square$-construction to assembler $K$-theory
  8. Examples
  9. Assemblers
  10. Varieties
  11. Definable sets
  12. Polytopes
  13. Application: a derived Euler characteristic

Key Result

Theorem A

Let $\mathcal{A}$ be an assembler satisfying certain axioms. Then there exists a squares category $\mathcal{C}^\textup{min}$ such that $K^\square(\mathcal{C}^\textup{min}) \simeq K(\mathcal{A})$.

Theorems & Definitions (83)

  • Theorem A: Proposition \ref{['prop:artificial_squares_cat_works']}
  • Proposition B: Proposition \ref{['prop:Tplus_vs_T']}
  • Proposition C: Proposition \ref{['prop:assmebler_K_th_is_squares_K_th']}
  • Proposition D: Corollary \ref{['cor:K_square_def_vs_K_pCGW_def']} and Proposition \ref{['prop:squares_cat_CS_agrees']}
  • Proposition E: Corollary \ref{['cor:map_of_spectra_def_ch']} and Proposition \ref{['prop:map_lifts_eulerchar']}
  • Definition 1.1: squares
  • Example 1.2: squares
  • Definition 1.3: squares
  • Definition 1.4: squares
  • Definition 1.5
  • ...and 73 more