[@humavit] we deduce that ${{\underline{Ht}}}$ is Grothendieck abelian whenever $t$ is non-degenerate. The results and arguments of the section are closely related to the properties of localizing subcategories of triangulated categories as studied by A. Neeman, H. Krause and other authors; see Remark \[rtst2\](\[it6\]) below for an “explanation” of this similarity. In §\[skan\] we study dualities between triangulated categories and torsion pairs orthogonal with respect to them. Considering a “very simple” duality we prove that for certain weight structures inside the bounded derived category of coherent sheaves on a scheme $X$ that is proper over ${{\operatorname{Spec}\,}}R$ (where $R$ is a noetherian ring) there necessarily exist (right or left) orthogonal $t$-structures.

Our main tools for constructing “more complicated” dualities are Kan extensions of (co)homological functors from a triangulated subcategory ${\underline{C}_0}$ to ${\underline{C}}$; their properties are rather interesting for themselves. We describe a duality between the homotopy category of filtered pro-objects for a stable proper Quillen model category ${\mathcal{M}}$ with ${\operatorname{Ho}}({\mathcal{M}})$. The properties of this duality imply that for any compactly generated $t$-structure on ${\operatorname{Ho}}({\mathcal{M}})$ its heart is a Grothendieck abelian category; they are also applied in [@bgn] to the study of [*motivic pro-spectra* ]{} and [*generalized coniveau spectral sequences*]{}.

For the 