We introduce the notion of noncompact (partial) silting and (partial) tilting sets and objects in any triangulated category D with arbitrary (set-indexed) coproducts. We show that equivalence classes of partial silting sets are in bijection with t-structures generated by their co-heart whose heart has a generator, and in case D is compactly generated, this restricts to: i) a bijection between equivalence classes of self-small partial silting objects and left nondegenerate t-structures in D whose heart is a module category and whose associated cohomological functor preserves products; ii) a bijection between equivalence classes of classical silting objects and nondegenerate smashing and co-smashing t-structures whose heart is a module category.We describe the objects in the aisle of the t-structure associated to a partial silting set T as Milnor (or homotopy) colimits of sequences of morphisms with successive cones in Sum(T )[n]. We use this fact to develop a theory of tilting objects in very general AB3 abelian categories, a setting and its dual in which we show the validity of several well-known results of tilting and cotilting theory of modules. Finally, we show that if T is a bounded tilting set in a compactly generated algebraic triangulated category D and H is the heart of the associated t-structure, then the inclusion H ֒→ D extends to a triangulated equivalence D(H) ∼ −→ D which restricts to bounded levels. * The authors thank Chrysostomos Psaroudakis, Jorge Vitória, Francesco Mattiello and Luisa Fiorot for their careful reading of two earlier versions of the paper and for their subsequent comments and suggestions which helped us a lot. We also thank JanŠťovíček for telling us about Lemma 7. Finally, the authors deeply thank the referee for the careful reading of the paper and for her/his comments and suggestions. Nicolás and Saorín are supported by research projects from the Spanish Ministerio de Economía y Competitividad (MTM2016-77445-P) and from the Fundación 'Séneca' of Murcia (19880/GERM/15), with a part of FEDER funds. Zvonareva is supported by the RFFI Grant 16-31-60089. The authors thank these institutions for their help. Zvonareva also thanks the University of Murcia for its hospitality during her visit, on which this research started.

We show that Quillen's small object argument works for exact categories under very mild conditions. This has immediate applications to cotorsion pairs and their relation to the existence of certain triangulated adjoint functors and model structures. In particular, the interplay of different exact structures on the category of complexes of quasi-coherent sheaves leads to a streamlined and generalized version of recent results obtained by Estrada, Gillespie, Guil Asensio, Hovey, Jørgensen, Neeman, Murfet, Prest, Trlifaj and possibly others.

We give a general parametrization of all the recollement data for a triangulated category with a set of generators. From this we deduce a characterization of when a ℵ 0 -perfectly generated (or aisled) triangulated category is a recollement of triangulated categories generated by a single compact object. Also, we use homological epimorphisms to give a complete and explicit description of all the recollement data for (or smashing subcategories of) the derived category of a k-flat dg category. In the final part we give a bijection between smashing subcategories of compactly generated triangulated categories and certain ideals of the subcategory of compact objects, in the spirit of H. Krause's work [Henning Krause, Cohomological quotients and smashing localizations, Amer. J. Math. 127 (2005Math. 127 ( ) 1191Math. 127 ( -1246. This bijection implies the following weak version of the generalized smashing conjecture: in a compactly generated triangulated category every smashing subcategory is generated by a set of Milnor colimits of compact objects.

We study the behavior of direct limits in the heart of a t-structure. We prove that, for any compactly generated t-structure in a triangulated category with coproducts, countable direct limits are exact in its heart. Then, for a given Grothendieck category G and a torsion pair t = (T , F) in G, we show that the heart Ht of the associated t-structure in the derived category D(G) is AB5 if, and only if, it is a Grothendieck category. If this is the case, then F is closed under taking direct limits. The reverse implication is true for a wide class of torsion pairs which include the hereditary ones, those for which T is a cogenerating class and those for which F is a generating class. The results allow to extend results by Buan-Krause and Colpi-Gregorio to the general context of Grothendieck categories and to improve some results of (co)tilting theory of modules.

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