Quasi DG categories and mixed motivic sheaves

We introduce the notion of a quasi DG category, generalizing that of a DG category. With a quasi DG category satisfying certain additional conditions, we associate another quasi DG category, the quasi DG category of C-diagrams. We then show the homotopy category of the quasi DG category of C-diagrams has the structure of a triangulated category. This procedure is then applied to produce a triangulated category of mixed motives over a base variety.