convenience of the reader we also make a list of the main definitions and notation used in this paper. Regular cardinals, Karoubi-closures and related matters, $D\perp E$, ${{}^{\perp}}D$, and $D{{}^{\perp}}$, suspended, cosuspended, extension-closed, and strict classes of objects, extension-closures, envelopes, subcategories generated by classes of objects of triangulated categories, homological and cohomological functors, compact objects, localizing and colocalizing subcategories, cogenerators, Hom-generators, compactly generated categories, cc, cp, and pp functors, and the Brown representability condition (along with its dual) are defined in §\[snotata\]; (smashing, cosmashing, countably smashing, adjacent, and compactly generated) torsion pairs (often denoted by $s=({\mathcal{LO}},{\mathcal{RO}})$; sometimes we also use the notation $(L_sM,R_sM)$), generators for them, and ${\mathcal{P}}$-null morphisms are introduced in §\[shop\]; $t$-structures ($t=({\underline{C}}^{t\le 0},{\underline{C}}^{t\ge 0})$) and several of their “types” (including smashing and cosmashing $t$-structures), their hearts (${{\underline{Ht}}}$), associated torsion pairs, $t$-homology ($H_0^t$), and $t$-projective objects are defined in §\[sts\]; weight structures ($w=({\underline{C}}_{w\le 0},{\underline{C}}_{w\ge 0})$; we also define ${\underline{C}}_{w=i}$ and ${\underline{C}}_{[m,n]}$), their “types”, hearts (${{\underline{Hw}}}$), and associated (weighty) torsion pairs, adjacent weight and $t$-structures, $m$-weight decompositions, negative subcategories, and the weight structure ${w^{st}}$ are introduced in §\[ssws\]; (weight) Postnikov towers along with the corresponding filtrations and (weight) complexes, weakly homotopic morphisms of complexes, and the category ${K_{\mathfrak{w}}}({{\underline{Hw}}})$ are 