homotopy categories. This matter is also related to the study of the stable homotopy category in [@prospect]. Let us now describe the contents of the paper. Some more information of this sort may be found in the beginnings of sections.

In §\[sold\] we we define torsion pairs and prove several of their properties; these are mostly simple but new. We also relate torsion pairs to $t$-structures and study $t$-projective objects (essentially following [@zvon]); they are related to adjacent weight structures that we will study later.

We start §\[sws\] from recalling some basics on weight structures (among those are some properties of weight complexes; though these are not really new, we treat this subject more accurately than in [@bws] where weight complexes were originally defined). Next we introduce pure functors (in §\[sdetect\]; their “construction” in Proposition \[ppure\] appears to be a very useful statement). An “intrinsic” definition of pure functors is given by Proposition \[pwrange\]. We relate the weight range of functors to virtual $t$-truncations. We also study the properties of virtual $t$-truncations and weight complexes under the assumption that $w$ is smashing. As an application, we treat the representability for virtual $t$-truncations of representable functors; this result has important applications 