Towards non-homotopy invariant motive theory,

In his fundamental work on triangulated category of mixed motives, Voevodsky developed a theory of homotopy invariant presheaves with transfers. We explain our attempt to generalize it to non-homotopy invariant theory. The basic idea is to replace homotopy invariance by Weil reciprocity. (Joint work with B. Kahn and S. Saito.)