AI 中文总结
该研究推广了g-向量扇概念，建立了0-奥斯兰德外三角范畴的厚子范畴与代数的τ-正交子范畴的双射，引入划分扇态射以统一相关范畴的函子关系。
AI 中文摘要
我们将g-向量扇的概念进行推广，使其适用于带有投射 silt 对象T的Hom-有限Krull-Schmidt 0-奥斯兰德k-线性外三角范畴𝒞。此外，我们证明该g-向量扇存在由厚子范畴诱导的、符合第二署名作者定义的容许划分。由此可定义𝒞的图范畴。我们建立了由包含所有投射-内射对象的预silt对象生成的𝒞的厚子范畴与T的自同态k-代数的τ-正交子范畴之间的双射，这表明我们的构造统一了此前所有图范畴与有限维代数的τ-簇态射范畴的构造。我们引入划分扇的态射，为不同代数与范畴的图范畴之间的函子关系提供统一框架。
英文摘要
We extend the notion of $\mathbf{g}$-vector fan so that it is defined for a Hom-finite Krull-Schmidt 0-Auslander $k$-linear extriangulated category $\mathcal{C}$ with a projective silting object $T$. Moreover, we show that the $\mathbf{g}$-vector fan admits an admissible partition, in the sense of the second-named author, which is induced by thick subcategories. One can thus define the picture category of $\mathcal{C}$. We establish a bijection between thick subcategories of $\mathcal{C}$ generated by presilting objects containing all projective-injective objects and $τ$-perpendicular subcategories of the endomorphism $k$-algebra of $T$. This shows that our construction unifies all previous constructions of picture categories and $τ$-cluster morphism categories of finite-dimensional algebras. We introduce morphisms of partitioned fans to provide a common framework for the functorial relationships between picture categories of different algebras and categories.
Comments44 pages, comments welcome!
