Purity in compactly generated derivators and t-structures with Grothendieck hearts

Abstract: We study t-structures with Grothendieck hearts on compactly generated triangulated categories $\mathcal{T}$ that are underlying categories of strong and stable derivators. This setting includes all algebraic compactly generated triangulated categories. We give an intrinsic characterisation of pure triangles and the definable subcategories of $\mathcal{T}$ in terms of directed homotopy colimits. For a left nondegenerate t-structure ${\bf t}=(\mathcal{U},\mathcal{V})$ on $\mathcal{T}$, we show that $\mathcal{V}$ is definable if and only if ${\bf t}$ is smashing and has a Grothendieck heart. Moreover, these conditions are equivalent to ${\bf t}$ being homotopically smashing and to ${\bf t}$ being cogenerated by a pure-injective partial cosilting object. Finally, we show that finiteness conditions on the heart of ${\bf t}$ are determined by purity conditions on the associated partial cosilting object.