A. Suslin and V. Voevodsky have realized a triangulated category which is supposed to be the bounded derived category of the hypothetical abelian category of mixed motives over , predicted by Grothendieck–Beilinson–Deligne. There are variants developed by Hanamura and M. Levine. There is a different “derived” approach to mixed motives, namely the -homotopy theory of F. Morel and V. Voevodsky.
See at motive the section Contructions of the derived category of mixed motives.
A. Beilinson, V. Vologodsky, A DG guide to Voevodsky’s motives, math.AG/0604004
M.V. Bondarko, Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura, math.AG/0601713
M.V. Bondarko, Weight structures and motives; comotives, coniveau and Chow-weight spectral sequences: a survey, arxiv:0903.0091