  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.

Representation Theory;Category Theory