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.
