defined in §\[sswc\]; pure homological functors ($H^{{\mathcal{A}}}$) and purely $R$-representable homology (with values in ${\operatorname{PShv}^R}({\underline{B}})$) is introduced in §\[sdetect\], virtual $t$-truncations ($\tau^{\ge m}H$ and $\tau^{\le m}H$), weight range of functors, and pure cohomological functors ($H_{{\mathcal{A}}}$) are defined in §\[svtt\]; functors of $R$-finite type and $R$-saturated categories are introduced in §\[sadjt\]; ${\alpha}$-small objects, countably perfect and perfect classes of objects, perfectly generated and well generated categories, weakly symmetric and symmetric classes, and Brown-Comenetz duals of functors and objects are defined in §\[scomp\]; countable homotopy colimits $\operatorname{\varinjlim}Y_i$, strongly extension-closed classes and strong extension-closures, (naive) big hulls, and zero classes of (collections of) functors are introduced in §\[scoulim\]; perfectly generated weight structures, ${\mathcal{P}}$-approximations, and contravariantly finite classes are defined in §\[sperfws\]; weakly and strongly ${\beta}$-well generated weight structures (and torsion pairs) are studied in §\[swgws\]; coextended functors and coextensions are defined in §\[scoext\]; (nice) dualities and orthogonal structures are introduced in §\[sdual1\]; biextensions are defined in §\[sdual1\]; stalk functors are introduced in §\[sgengroth\]; certain model structures on pro-objects are recalled in §\[sprospectra\].

The author is deeply grateful to prof. F. Déglise, prof. G.C. Modoi, prof. Salorio M.J. Souto, and prof. J. Šťovíček for their very useful comments.

On torsion pairs and $t$-structures (“simple 