# A $2$-torsion version of the motivic stable homotopy category?

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.

• Why would you want this? – Will Sawin Dec 7 '13 at 18:11
• Because I want certain '$2$-torsion motivic spectra' for smooth varieties. – Mikhail Bondarko Dec 7 '13 at 18:14
• What do you mean by $2$-torsion? $2\cdot\operatorname{id}_X=0$ for any object $X$? – Fernando Muro Dec 7 '13 at 22:11
• 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). – Mikhail Bondarko Dec 7 '13 at 22:26
• There are semi-trivial 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 $X\rightarrow ip(X)$. – Fernando Muro Dec 7 '13 at 23:47