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 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.

Using techniques due to Dwyer-Greenlees-Iyengar we construct weight structures in triangulated categories generated by compact objects. We apply our result to show that, for a dg category whose homology vanishes in negative degrees and is semi-simple in degree 0, each simple module over the homology lifts to a dg module which is unique up to isomorphism in the derived category. This allows us, in certain situations, to deduce the existence of a canonical t-structure on the perfect derived category of a dg algebra. From this, we can obtain a bijection between hearts of t-structures and sets of so-called simple-minded objects for some dg algebras (including Ginzburg algebras associated to quivers with potentials). In three appendices, we elucidate the relation between Milnor colimits and homotopy colimits and clarify the construction of t-structures from sets of compact objects in triangulated categories as well as the construction of a canonical weight structure on the unbonded derived category of a non positive dg category.

Communicated by I. ReitenMSC:a b s t r a c t Curved A ∞ -algebras appear in nature as deformations of dg algebras. We develop the basic theory of curved A ∞ -algebras and, in particular, curved dg algebras. We investigate their link with a suitable class of dg coalgebras via the bar construction and produce Quillen model structures on their module categories. We define the analogue of the relative derived category for a curved dg algebra.

a b s t r a c tSince curved dg algebras, and modules over them, have differentials whose square is not zero, these objects have no cohomology, and there is no classical derived category. For different purposes, different notions of ''derived'' categories have been introduced in the literature. In this article, we show that for some concrete curved dg algebras, these derived categories vanish. This happens for example for the initial curved dg algebra whose module category is the category of precomplexes, and for certain deformations of dg algebras.