Silting Theory in triangulated categories with coproducts

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 bijection restricts to one between equivalence classes of self-small partially silting objects and left nondegenerate t-structures in D whose heart is a module category and whose associated cohomological functor preserves products. We describe the objects in the aisle of the t-structure associated to a partial silting set T as the Milnor (aka homotopy) colimit of sequences of morphisms with succesive 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 on 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 of H in D extends to a triangulated equivalence between the derived category D(H) of H and the ambient triangulated category D which restricts to bounded levels.