  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.
