Silting reduction and picture categories of 0-Auslander extriangulated categories

TL;DR

This paper extends silting reduction techniques to 0-Auslander extriangulated categories, introduces the picture category concept, and explores their algebraic and categorical properties, including conditions for quotients and fundamental groups.

Contribution

It generalizes Iyama--Yang silting reduction to extriangulated categories, defines the picture category for 0-Auslander categories, and analyzes their algebraic structures.

Findings

Verdier quotient admits an extriangulation under condition (gCP)

The quotient remains 0-Auslander when the condition holds

The picture group is finitely presented when H_0 is g-finite

Abstract

Let $\mathcal{C}$ be an extriangulated category and let $\mathcal{R}\subseteq \mathcal{C}$ be a rigid subcategory. Generalizing Iyama--Yang silting reduction, we devise a technical condition $\textbf{(gCP)}$ on $\mathcal{R}$ which is sufficient for the Verdier quotient $\mathcal{C}/\mathrm{thick}(\mathcal{R})$ to be equivalent to an ideal quotient. In particular, the Verdier quotient $\mathcal{C}/\mathrm{thick}(\mathcal{R})$ will admit an extriangulation in such a way that the localization functor $L_{\mathcal{R}}\colon \mathcal{C} \rightarrow \mathcal{C}/\mathrm{thick}(\mathcal{R})$ is extriangulated. When $\mathcal{C}$ is 0-Auslander, the condition $\textbf{(gCP)}$ holds for all rigid subcategories $\mathcal{R}$ admitting Bongartz completions. Furthermore, we prove that the Verdier quotient $\mathcal{C}/\mathrm{thick}(\mathcal{R})$ then remains 0-Auslander. As an application, we define the picture category of a connective $0$-Auslander exact dg category $\mathscr{A}$ with Bongartz completions, which generalizes the notion of $\tau$-cluster morphism category. We show that the picture category of $\mathscr{A}$ is a cubical category, in the sense of Igusa. The picture group of $\mathscr{A}$ is defined as the fundamental group of its picture category. When $H_0\mathscr{A}$ is $\mathbf{g}$-finite, the picture group of $\mathscr{A}$ is finitely presented.