$g$-vector fans and picture categories for 0-Auslander extriangulated categories

Abstract: 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.