Grothendieck has envisioned a deep picture on a hypothetical category of pure and mixed motives of varieties which has partly been fullfilled so far. This entry gives only some pointers to approaches via dg-categories and their cousins, pretriangulated -categories.
Using derived algebraic geometry Maxim Kontsevich embeds the search for motives of varieties into the search of motives of the dg or -enhancements of the derived categories of coherent sheaves on the noncommutative varieties, what denotes in algebraic geometry working with a bit larger class of pretriangulated -categories. There are also dg-approaches to Voevodsky’s derived category of mixed motive?s.
online video of M.K. talk at the conference on 61st birthday of Deligne, at IAS Princeton.
M.V. Bondarko, Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura, arXiv:math.AG/0601713
A. Beilinson, V. Vologodsky, A DG guide to Voevodsky’s motives, arXiv:math.KT/0604004
S. Mahanta, Noncommutative geometry in the framework of differential graded categories, arXiv:0805.1628
Maxim Kontsevich, Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435