  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.~To\"en and M.~Vaqui\'e, 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.
