We define and study transversal weight and t-structures (for triangulated categories); if a weight structure is transversal to a t-one, then it defines certain `weights' for its heart. Our results axiomatize and describe in detail the relations between the Chow weight structure wChow for Voevodsky's motives (introduced in a preceding paper), the (conjectural) motivic t-structure, and the conjectural weight filtration for them. This picture becomes non-conjectural when restricted to the derived categories of Deligne's 1-motives (over a smooth base) and of Artin-Tate motives over number fields. In particular, we prove that the `weights' for Voevodsky's motives (that are given by wChow) are compatible with those for 1-motives (that were `classically' defined using a quite distinct method); this result is new. Weight structures transversal to the canonical t-structures also exist for the Beilinson's Db ~Hp (the derived category of graded polarizable mixed Hodge complexes) and for the derived category of (Saito's) mixed Hodge modules.
We also study weight filtrations for the heart of t and (the degeneration of) weight spectral sequences. The corresponding relation between t and w is strictly weaker than transversality; yet it is easier to check, and we still obtain a certain filtration for (objects of) the heart of t that is strictly respected by morphisms.
In a succeeding paper we apply the results obtained in order to reduce the existence of Beilinson's mixed motivic sheaves (over a base scheme S) and `weights' for them to (certain) standard motivic conjectures over a universal domain K.
Homology, Homotopy and Applications, Vol. 14 (2012), No. 1, pp.239-261.
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