Homotopy types and geometries below Spec(ℤ)

Abstract

After the first heuristic ideas about "the field of one element" F 1 and "geometry in characteristics 1" (J. Tits, C. Deninger, M. Kapranov, A. Smirnov et al.), there were developed several general approaches to the construction of "geometries below Spec Z". Homotopy theory and the "the brave new algebra" were taking more and more important places in these developments, systematically explored by B. Toën and M. Vaquié, among others.This article contains a brief survey and some new results on counting problems in this context, including various approaches to zeta-functions and generalised scissors congruences.

Quantum Statistical Mechanics of the Absolute Galois Group

“…In [48] and [54] it was shown that the endomorphisms σ n and ρ n of the Bost-Connes algebra lift to various equivariant Grothendieck rings and further to the level of assemblers and homotopy theoretic spectra. The symmetries of the Bost-Connes system, given byẐ * in the original version, or by mGT according to Proposition 3.2 above, also can be lifted to these categorical and homotopy theoretic levels in a similar manner.…”