Bost–Connes Systems and F1-structures in Grothendieck Rings, Spectra, and Nori Motives

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 F 1 -geometry, in the framework of torifications, that fit into this general setting.