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.

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.

In this paper, we generally describe a method of taking an abstract six functors formalism in the sense of Khan or Cisinski-Déglise, and outputting a derived motivic measure in the sense of Campbell-Wolfson-Zakharevich. In particular, we use this framework to define a lifting of the Gillet-Soulé motivic measure.Conventions: Throughout this paper, k will always refer to a perfect field, and R will refer to an arbitrary commutative ring. Furthermore, given a base scheme S, the category Var S of varieties over S will simply be the category of finite type separated schemes over S (we do not require our varieties to be reduced). Importantly, unless otherwise stated, our ambient ∞-cosmos [38] will be that of Kan complex-enriched categories, the so-called fibrant S-categories of . This is because we will be working with commutative diagrams directly for much of the work, and it is easier to prove commutativity in a model with strict horizontal composition. That said, we may invoke comparison between the K-theory of S-categories and that of Quasicategories developed in [8] .

In this note, we provide an explicit non-Quillen equivalence between the category of precubical sets and Gaucher's category of flows via a class of "realization functors" (with mild assumptions on the cofibrations of the category of precubical sets). In addition, we demonstrate a Quillen equivalence between simplicial semicategories and flows before proving that simplicial semicategories satisfy many of the same properties as flows. Finally, we introduce the category of boxed symmetric trees, presheaves on which may provide a slightly more flexible setting for concurrent computing than (pre)cubical sets, before showing that when endowed with degeneracies, the aforementioned presheaf category is a test category (although not strict test).