We study t-structures with Grothendieck hearts on compactly generated triangulated categories $${\mathcal {T}}$$ 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 $${\mathcal {T}}$$ T in terms of directed homotopy colimits. For a left nondegenerate t-structure $$\mathbf{t}=({\mathcal {U}},{\mathcal {V}})$$ t = ( U , V ) on $${\mathcal {T}}$$ T , we show that $${\mathcal {V}}$$ V is definable if and only if $$\mathbf{t}$$ t is smashing and has a Grothendieck heart. Moreover, these conditions are equivalent to $$\mathbf{t}$$ t being homotopically smashing and to $$\mathbf{t}$$ t being cogenerated by a pure-injective partial cosilting object. Finally, we show that finiteness conditions on the heart of $$\mathbf{t}$$ t are determined by purity conditions on the associated partial cosilting object.

中文翻译：

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格洛腾迪克心的紧凑生成的导数和 t 结构的纯度

我们在紧凑生成的三角化类别 $${\mathcal {T}}$$ T 上使用格罗腾迪克心研究 t 结构，这些类别是强和稳定导数的潜在类别。此设置包括所有代数紧凑生成的三角化类别。我们根据有向同伦共限给出了纯三角形和 $${\mathcal {T}}$$ T 的可定义子类别的内在特征。对于左非退化 t 结构 $$\mathbf{t}=({\mathcal {U}},{\mathcal {V}})$$ t = ( U , V ) 在 $${\mathcal {T} }$$ T ，我们证明 $${\mathcal {V}}$$ V 是可定义的，当且仅当 $$\mathbf{t}$$ t 正在粉碎并且具有 Grothendieck 心脏。此外，这些条件等价于 $$\mathbf{t}$$ t 同调粉碎和 $$\mathbf{t}$$ t 由纯注入部分共混对象共同生成。最后，