Presilting sequences for 0-Auslander extriangulated categories

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) $τ$-exceptional sequences over $Λ= \text{End}_{\mathscr{C}}(P)$, where $P$ is a projective generator of $\mathscr{C}$. This correspondence provides a new perspective on the Buan--Marsh bijection between signed $τ$-exceptional sequences and ordered support $τ$-rigid objects. Furthermore, we introduce a new category $\mathfrak{M}(\mathscr{C})$, called the $τ$-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 $τ$-cluster morphism category of $Λ$ from $\mathfrak{M}(\mathscr{C})$.