For a field $k$ there exists the motivic stable homotopy $SH(k)$; it is compactly generated. My question: does there exist a 'reasonable' functor $p$ from $SH(k)$ to a certain triangulated category $SH_2(k)$ that posesses a right adjoint $i$ such that any object of $ip(SH(k))$ is $2$-torsion or (at least) a countable homotopy colimit of $2$-torsion objects? As far as I remember, in a paper of Schwede it is proved that nothing like $SH(k)\otimes \mathbb{Z}/2^n\mathbb{Z}$ can exist; is it possible to 'project' $SH(k)$ onto something like the subcategory of ind-$2$-torsion objects? I don't know what I want exactly; yet this should be some 'universal' construction.

Here is one answer. Start with the 2-local stable homotopy category. Map X to the fiber C_0 X of the map X --> L_0 X to the rationalization of X. Then C_0 X is the minimal weak colimit of all the finite torsion spectra mapping to X, so it is an ind-2-torsion object. It also remembers, by definition, all the information about X except the rational information. The category consisting of all the C_0 X is a full triangulated subcategory of the 2-local stable homotopy category closed under coproducts and the mod 2 Moore spectrum is a weak generator. It is closed under the smash product, but it is not symmetric monoidal because there is no unit. The image of the sphere looks just like the sphere except that the 0th stable homotopy group is 0 and in dimension -1 you see the Prufer group Q/Z_(2).

1more comment