Toen and Vaquie: Under Spec Z. Some notes: Idea: Relative alg geom. Think of commutative monoids in a symm monoidal cat C as models for affine schemes relative to C. If there is a reasonable symmetric monoidal functor from C to Z-modules, get a base change functor, and a notion of scheme under Spec(Z). Homotopical version of this requires C to have a model structure. Now have flat and Zariski topology. Can make sense of schemes: a functor with a Zariski covering. Stuff about toric varieties and GL. Brave new AG over the sphere spectrum, and the spectrum with one element. Digression: Flat and Zariski ok. For etale (and maybe hence Nisnevich), Peter Arndt said there might be three ways of doing it: by a lifting property, by factorization systems (Anel), or by mimicking something Deitmar does for monoids, see Peter’s thesis in progress, and see also the notion of formally etale.
nLab page on Relative algebraic geometry