Let T {\mathcal{T}} be a triangulated category with a proper class ξ \xi of triangles and X {\mathcal{X}} be a subcategory of T {\mathcal{T}} . We first introduce the notion of X {\mathcal{X}} -resolution dimensions for a resolving subcategory of T {\mathcal{T}} and then give some descriptions of objects having finite X {\mathcal{X}} -resolution dimensions. In particular, we obtain Auslander-Buchweitz approximations for these objects. As applications, we construct adjoint pairs for two kinds of inclusion functors and characterize objects having finite X {\mathcal{X}} -resolution dimensions in terms of a notion of ξ \xi -cellular towers. We also construct a new resolving subcategory from a given resolving subcategory and reformulate some known results.

中文翻译：

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解决三角分类中的分辨率维度

令T {\ mathcal {T}}为三角形的类别，具有三角形的正确类别ξ\ xi，而X {\ mathcal {X}}为T {\ mathcal {T}}的子类别。我们首先介绍X {\ mathcal {X}}分辨率维度的概念，以解决T {\ mathcal {T}}的子类别，然后对具有有限X {\ mathcal {X}}分辨率维度的对象进行一些描述。特别是，我们获得了这些物体的Auslander-Buchweitz近似值。作为应用程序，我们为两种包含函子构造了伴随对，并根据ξ\ xi-细胞塔的概念来表征具有有限X {\ mathcal {X}}-分辨率尺寸的对象。我们还根据给定的解决方案子类别构造了一个新的解决方案子类别，并重新制定了一些已知的结果。