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.

$\begingroup$ Why would you want this? $\endgroup$ – Will Sawin Dec 7 '13 at 18:11

$\begingroup$ Because I want certain '$2$torsion motivic spectra' for smooth varieties. $\endgroup$ – Mikhail Bondarko Dec 7 '13 at 18:14

$\begingroup$ What do you mean by $2$torsion? $2\cdot\operatorname{id}_X=0$ for any object $X$? $\endgroup$ – Fernando Muro Dec 7 '13 at 22:11

$\begingroup$ Rather $2^N id_X=0$ for some $N$ (that could be fixed or vary with $X$; the case $N=1$ could also be interesting). $\endgroup$ – Mikhail Bondarko Dec 7 '13 at 22:26

$\begingroup$ There are semitrivial solutions, like smashing with a ring spectrum where $2^N$ vanishes, e.g.~an EML spectrum of a ring where $2^N=0$. Maybe you want to reformulate your question imposing more conditions, say on the unit $X\rightarrow ip(X)$. $\endgroup$ – Fernando Muro Dec 7 '13 at 23:47
Here is one answer. Start with the 2local 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 ind2torsion 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 2local 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).