of compactly generated ones (in Theorem \[tclass\]); we drop the assumption that ${\underline{C}}$ is a “stable derivator” category that was necessary for the proof of the closely related Corollary 3.8 of [@postov]. We also relate [*adjacent*]{} torsion pairs to “Brown-Comenetz-type symmetry”.

We also study [*dualities*]{} between triangulated categories and their relation to torsion pairs. We demonstrate that for a $t$-structure $t$ on ${\underline{C}'}$ that does not possess a left adjacent weight structure there still may exist $w$ on a category ${\underline{C}}$ that is [$\Phi$-orthogonal ]{} to $t$, where $\Phi:{\underline{C}}{{^{op}}}\times {\underline{C}}'\to {\underline{\operatorname{Ab}}}$ is a (“bicontinuous”) duality of triangulated categories. Moreover, for an interesting sort of dualities that we study in §\[sdual2\] and for any compactly generated $t$ on ${\underline{C}'}$ the objects of the heart of the corresponding orthogonal $w$ give faithful exact “stalk” functors ${{\underline{Ht}}}\to {\underline{\operatorname{Ab}}}$ that respect coproducts; taking the functor $\Phi(P,-)$ for $P$ being a [*cogenerator*]{} of ${{\underline{Hw}}}$ we easily obtain that ${{\underline{Ht}}}$ is an AB5 abelian category (cf. Theorem \[tgroth\]). So we demonstrate once again that weight structures “shed some light on $t$-structures”.

Moreover, dualities are important for [@bgn] where they are applied to the study of coniveau spectral sequences and the homotopy $t$-structures on various motivic stable 