| Summary: | Let <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="script">C</mi><mo>,</mo><mi mathvariant="double-struck">E</mi><mo>,</mo><mi mathvariant="fraktur">s</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> be an extriangulated category with a proper class <inline-formula><math display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> of <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">E</mi></semantics></math></inline-formula>-triangles and <inline-formula><math display="inline"><semantics><mi mathvariant="fraktur">X</mi></semantics></math></inline-formula> a resolving subcategory of <inline-formula><math display="inline"><semantics><mi mathvariant="script">C</mi></semantics></math></inline-formula>. In this paper, we introduce the notion of <inline-formula><math display="inline"><semantics><mi mathvariant="fraktur">X</mi></semantics></math></inline-formula>-resolution dimension relative to the subcategory <inline-formula><math display="inline"><semantics><mi mathvariant="fraktur">X</mi></semantics></math></inline-formula> in <inline-formula><math display="inline"><semantics><mi mathvariant="script">C</mi></semantics></math></inline-formula>, and then give some descriptions of objects with finite <inline-formula><math display="inline"><semantics><mi mathvariant="fraktur">X</mi></semantics></math></inline-formula>-resolution dimension. In particular, we obtain Auslander-Buchweitz approximations for these objects. As applications, we construct adjoint pairs for two kinds of inclusion functors, and construct a new resolving subcategory from a given resolving subcategory which reformulates some known results.
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