1-motivic sheaves and the Albanese functor

Using sheaf theoretic methods, we define functors Lπ0:DMeff(k)→D(HI≤0(k)) and LAlb:DMeff(k)→D(HI≤1(k)). The functor LAlb extends the one in [L. Barbieri-Viale, B. Kahn, On the derived category of 1-motives, I. Prépublication Mathématique de l’IHÉS (M/07/22), June 2007, 144 pages] to non-necessarily geometric motives. These functors are then used to define higher Néron–Severi groups and higher Albanese sheaves.