For any $c>0$ does there exist an object $C$ of Voevodsky's $DM^{eff}_$ (over the field of complex numbers) such that for any $i\le c$ and any smooth variety $X$ the $i$th singular cohomology of $X$ is given by the $i$th hypercohomology of $C$ at $X$? It seems that I can prove this fact (using qfhdescent), but if somebody has proved this already I would prefer to have a reference.
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6$\begingroup$ You are right, the simplest is to use qfh descent in order to prove that Ayoub's spectrum in "Note sur les opérations de Grothendieck et la réalisation de Betti", J. Inst. Math. Jussieu, Volume 9 (2010), no. 2, 225263, is canonically endowed with transfers, whence belongs to $DM$. Also, if you drop the $$ in $DM^{eff}_{}$ (which is absolutely useless), and work in $DM^{eff}$ (i.e. $\mathbf{A}^1$localization of the (unbounded) derived category of Nisnevich sheaves with transfers), you get an object which represents Betti cohomology in any degree (giving up this strange $c>0$ of yours). $\endgroup$– D.C. CisinskiOct 18, 2010 at 21:59

$\begingroup$ In fact, I always wanted to know: is it absolutely safe to work in $DM^{eff}$?:) Thank you! $\endgroup$– Mikhail BondarkoOct 19, 2010 at 9:45

$\begingroup$ It seems that there is a paper by Florence Lecomte and Nathalie Wach which does exactly what you are asking for (they even deal with this famous $DM^{eff}_$,but the exactly same thing works for unbounded motivic complexes): see arXiv:0911.5611. $\endgroup$– D.C. CisinskiOct 20, 2010 at 14:32
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