On torsion pairs, (well generated) weight structures, adjacent $t$-structures, and related (co)homological functors

The paper contains a collection of results related to weight structures, $t$-structures, and (more generally) to torsion pairs. For any weight structure $w$ we study (co)homological pure functors; these "ignore all weights except weight zero" and have already found several applications. We also study virtual $t$-truncations of cohomological functors coming from $w$. These are closely related to $t$-structures; so we prove in several cases (including certain categories of coherent sheaves) that $w$ "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 weight structure). The existence of weight structures right adjacent to compactly generated $t$-structures (and constructed using Brown-Comenetz duality) implies that the hearts of the latter have injective cogenerators and satisfy the AB3* axiom; actually, "most of them" are Grothendieck abelian (due to the existence of "regularly 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 properties of torsion pairs (that are rather parallel to that of weight structures); we also generalize a theorem of D. Pospisil and J. Stovicek to obtain a classification of compactly generated torsion pairs.