It is a great pleasure for me to dedicate this paper to Andrei Alexandrovich Suslin on the occasion of his sixtieth birthday. M.V. Bondarko * March 21, 2016Abstract In this paper we introduce a new notion of weight structure (w) for a triangulated category C; this notion is an important natural counterpart of the notion of t-structure. It allows extending several results of the preceding paper [Bon09] to a large class of triangulated categories and functors.The heart of w is an additive category Hw ⊂ C. We prove that a weight structure yields Postnikov towers for any X ∈ ObjC (whose 'factors' X i ∈ ObjHw). For any (co)homological functor H : C → A (A is abelian) such a tower yields a weight spectral sequence T :T is canonical and functorial in X starting from E 2 . T specializes to the usual (Deligne) weight spectral sequences for 'classical' realizations of Voevodsky's motives DM ef f gm (if we consider w = w Chow with Hw = Chow ef f ) and to Atiyah-Hirzebruch spectral sequences in topology.We prove that there often exists an exact conservative weight complex functor C → K(Hw). This is a generalization of the functor(which is an extension of the weight complex of Gillet and Soulé). We prove that K 0 (C) ∼ = K 0 (Hw) under certain restrictions.

We describe explicitly the Voeovodsky's triangulated category of motives DM ef f gm (and describe a 'differential graded enhancement' for it). This enables us to able to verify that DM gm Q is (anti)isomorphic to Hanamura's D(k).We obtain a description of all subcategories (including those of Tate motives) and of all localizations of DM

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The main goal of this paper is to define the Chow weight structure w Chow for the category DM c (S) of (constructible) Beilinson motives over any excellent separated finite-dimensional base scheme S (this is the version of Voevodsky's motives over S described by Cisinski and Deglise). We also study the functoriality properties of w Chow (they are very similar to those for weights of mixed complexes of sheaves, as described in §5 of [BBD82]).As shown in a preceding paper, (the existence of) w Chow automatically yields a certain exact conservative weight complex functor DM c (S) → K b (Chow(S)). Here Chow(S) is the heart of w Chow ; it is 'generated' by motives of regular schemes that are projective over S. We also obtain that K 0 (DM c (S)) ∼ = K 0 (Chow(S)) (and define a certain 'motivic Euler characteristic' for S-schemes).Besides, we obtain (Chow)-weight spectral sequences and filtrations for any (co)homology of motives; we discuss their relation to Beilinson's 'integral part' of motivic cohomology and with weights of mixed complexes of sheaves. For the study of the latter we also introduce a new formalism of relative weight structures.

International audienceThe aim of this work is to construct certain homotopy t-structures on various categories of motivic homotopy theory, extending works of Voevodsky, Morel, Déglise and Ayoub. We prove these t-structures possess many good properties, some analogous to those of the perverse t-structure of Beilinson, Bernstein and Deligne. We compute the homology of certain motives, notably in the case of relative curves. We also show that the hearts of these t-structures provide convenient extensions of the theory of homotopy invariant sheaves with transfers, extending some of the main results of Voevodsky. These t-structures are closely related to Gersten weight structures as defined by Bondarko