# Category of motivic spectra

When the survey Axiomatic Stable Homotopy, Neil Strickland, 2004 was written the category of motivic spectra was not investigated from the point of view of axiomatic stable homotopy, as considered e.g. in Hovey, Palmieri and Strickland, Axiomatic Stable Homotopy Theory, 1997.

The question is this: was any work done in the last ten years in this direction?

• Dear @Andrei Halanay: The first link you provide appears to be inaccessible without a mathscinet subscription. – Ricardo Andrade Oct 18 '13 at 20:41
• @RicardoAndrade: details of linked references now edited in. – Peter LeFanu Lumsdaine Oct 18 '13 at 22:12
• The survey "Axiomatic stable homotopy" by Strickland is available in the arXiv: arxiv.org/abs/math/0307143 – Ricardo Andrade Oct 18 '13 at 22:19

ETA In case someone stumbles upon this answer, let me take back one cent and say that axiom (c) should not hold over any scheme of positive dimension. The correct expectation is that it holds over zero-dimensional schemes (this being only known in characteristic zero for now). The idea is that the subcategory generated by dualizable object is the category of locally constant motivic spectra. Here is a proof that (c) does not hold for $SH(S)$ assuming $S$ admits a regular codimension 1 point $s$ (eg $S$ is normal): take an open $U$ whose restriction to $\mathrm{Spec}(\mathcal{O}_{S,s})$ is the generic point. Then $\Sigma^\infty_T U_+\in SH(S)$ is not generated by dualizable objects; in fact its rational motive isn't: localization and absolute purity imply easily that pullback to the generic point $DM_{\mathbb Q}(\mathcal{O}_{S,s})\to DM_{\mathbb Q}(\eta)$ is conservative on locally constant motivic sheaves.