# Bost-Connes systems and F1-structures in Grothendieck rings, spectra, and Nori motives

## TL;DR

This paper develops geometric and categorical frameworks for Bost-Connes algebras within Grothendieck rings, spectra, and Nori motives, connecting algebraic, motivic, and F1-geometry concepts.

## Contribution

It introduces new categorifications of Bost-Connes algebras using Grothendieck rings, spectra, and Nori motives, linking them through Euler characteristics and zeta functions.

## Key findings

- Constructed geometric lifts of Bost-Connes algebra to Grothendieck rings and spectra.
- Established relations between categorifications and algebra via Euler characteristics and zeta functions.
- Discussed F1-geometry within the context of torifications and motivic frameworks.

## Abstract

We construct geometric lifts of the Bost-Connes algebra to Grothendieck rings and to the associated assembler categories and spectra, as well as to certain categories of Nori motives. These categorifications are related to the integral Bost-Connes algebra via suitable Euler characteristic type maps and zeta functions, and in the motivic case via fiber functors. We also discuss aspects of F1-geometry, in the framework of torifications, that fit into this general setting.