Presilting sequences for 0-Auslander extriangulated categories

TL;DR

This paper introduces presilting sequences in 0-Auslander extriangulated categories, establishing a bijection with tau-exceptional sequences and defining a new tau-cluster morphism category, extending existing bijections and categories.

Contribution

It defines presilting sequences in extriangulated categories, links them to tau-exceptional sequences, and introduces the tau-cluster morphism category, broadening the theoretical framework.

Findings

Bijection between presilting sequences and tau-exceptional sequences.

Introduction of the tau-cluster morphism category of a category.

Recovery of the tau-cluster morphism category of an algebra from the new category.

Abstract

Let $\mathscr{C}$ be a reduced $0$-Auslander extriangulated category. Motivated by Pan--Zhu silting reduction for such categories, we introduce the notion of (signed) presilting sequences in $\mathscr{C}$ and establish a bijection between (signed) presilting sequences in $\mathscr{C}$ and (signed) $\tau$-exceptional sequences over $\Lambda = \text{End}_{\mathscr{C}}(P)$, where $P$ is a projective generator of $\mathscr{C}$. This correspondence provides a new perspective on the Buan--Marsh bijection between signed $\tau$-exceptional sequences and ordered support $\tau$-rigid objects. Furthermore, we introduce a new category $\mathfrak{M}(\mathscr{C})$, called the $\tau$-cluster morphism category of $\mathscr{C}$, whose objects are certain extension-closed subcategories of $\mathscr{C}$ and whose morphisms are described in terms of signed presilting sequences. As an application, we recover the $\tau$-cluster morphism category of $\Lambda$ from $\mathfrak{M}(\mathscr{C})$.