Homotopy types and geometries below Spec(ℤ)

After the first heuristic ideas about 'the field of one element' F₁ 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. The new version includes considerable extensions and revisions suggested by I. Zakharevich