All K-theory is squares K-theory

TL;DR

This paper demonstrates that the K-theory spectra of various assemblers, including geometric and definable set assemblers, are equivalent to the K-theory spectrum of a squares category, enabling new connections with Euler characteristics.

Contribution

It establishes a unifying framework showing all these K-theories are equivalent to squares K-theory, extending the understanding of assembler spectra.

Findings

K-theory spectra of many assemblers are equivalent to squares K-theory

Lifted definable Euler characteristic to a map of K-theory spectra

Unified various geometric and definable set assemblers under a common K-theory framework

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.