Quasi-resolving subcategories and dimensions in extriangulated categories

Let $\mathcal{C}=(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. In this paper, we introduce and study quasi-resolving subcategories in $\mathcal{C}$. More precisely, we first introduce the notion of $\mathcal{X}$-resolution dimensions for a quasi-resolving subcategory $\mathcal{X}$ of $\mathcal{C}$ and then give some equivalent characterizations of objects which have finite $\mathcal{X}$-resolution dimensions. As an application, we introduce Gorenstein quasi-resolving subcategories, denoted by ${\mathcal{G}\mathcal{Q}\mathcal{P}}_{\mathcal{X}}(\xi )$, in term of a quasi-resolving subcategory $\mathcal{X}$, and prove that ${\mathcal{G}\mathcal{Q}\mathcal{P}}_{\mathcal{X}}(\xi )$ is also a quasi-resolving subcategory of $\mathcal{C}$. Moreover, some classical known results are generalized in ${\mathcal{G}\mathcal{Q}\mathcal{P}}_{\mathcal{X}}(\xi )$.