Discrete derived categories II: The silting pairs CW complex and the stability manifold

TL;DR

This paper introduces a CW complex of silting pairs for discrete derived categories, proves its contractibility, and embeds it into the stability manifold, showing the stability space is contractible.

Contribution

It defines the silting pairs CW complex for triangulated categories and demonstrates its contractibility in discrete derived categories, linking it to the stability manifold.

Findings

The silting pairs CW complex is contractible for discrete derived categories.

An explicit embedding from the CW complex into the stability manifold is provided.

The space of stability conditions for discrete derived categories is shown to be contractible.

Abstract

Discrete derived categories were studied initially by Vossieck \cite{Vossieck} and later by Bobi\'nski, Gei\ss, Skowro\'nski \cite{BGS}. In this article, we define the CW complex of silting pairs for a triangulated category and show that it is contractible in the case of discrete derived categories. We provide an explicit embedding from the silting CW complex into the stability manifold. By work of Qiu and Woolf, there is a deformation retract of the stability manifold onto the silting pairs CW complex. We obtain that the space of stability conditions of discrete derived categories is contractible.