Simple-minded systems and reduction for negative Calabi-Yau triangulated categories

TL;DR

This paper introduces a reduction technique for negative Calabi-Yau triangulated categories and explores the properties of simple-minded systems, revealing parallels with cluster-tilting objects and offering an inductive construction method.

Contribution

It develops the theory of w-simple-minded systems in negative Calabi-Yau categories and provides a novel reduction technique and inductive construction method.

Findings

Establishes basic properties of w-simple-minded systems.

Provides a reduction technique for negative Calabi-Yau categories.

Shows parallels between simple-minded systems and cluster-tilting objects.

Abstract

We develop the basic properties of $w$-simple-minded systems in $(-w)$-Calabi-Yau triangulated categories for $w \geq 1$. The main result is a reduction technique for negative Calabi-Yau triangulated categories. We show that the theory of simple-minded systems exhibits striking parallels with that of cluster-tilting objects. Our construction provides an inductive technique for constructing simple-minded systems.