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abstract: |
    This paper contains a rich collection of results related to weight structures and $t$-structures. For any weight structure $w$ we study [*pure*]{} (co)homological functors; these “ignore all weights except weight zero” and have already found several applications (in particular, to Picard groups of triangulated categories). Moreover, we study [*virtual $t$-truncations*]{} of cohomological functors. The resulting functors are defined in terms of $w$ but are closely related to $t$-structures; so we prove in several cases that a weight structure $w$ “gives” a $t$-structure (that is [*adjacent*]{} or [*$\Phi$-orthogonal*]{} to it).

    We also study in detail [*well generated*]{} weight structures (and prove that any [ *perfect*]{} set of objects generates a certain weight structure). We prove the existence of weight structures right adjacent to [*compactly generated*]{} $t$-structures (using Brown-Comenetz duality); this implies that the hearts of the latter have injective cogenerators and satisfy the AB3\* axiom. Actually, “most of” these hearts are Grothendieck abelian (due to the existence of “bicontinuously orthogonal” weight structures).

    It is convenient for us to use the notion of [*torsion pairs*]{}; these essentially generalize both weight structures and $t$-structures. We prove several new properties of torsion pairs; in particular, we generalize a theorem of D. 