2-Calabi-Yau categories with a directed cluster-tilting subcategory

TL;DR

This paper extends the theory of 2-Calabi-Yau categories by exploring those with directed cluster-tilting subcategories, revealing their cluster structures and connections to representations of thread quivers, and establishing a framework for understanding their cluster-tilting subcategories.

Contribution

It introduces the concept of 2-Calabi-Yau categories with directed cluster-tilting subcategories and analyzes their cluster structures and combinatorics, linking them to representations of thread quivers.

Findings

Categories have a cluster structure.

Cluster-tilting subcategories relate to triangulations of cyclically ordered sets.

Existence of a cluster map on all rigid objects in certain categories.

Abstract

As a generalization of acyclic 2-Calabi-Yau categories, we consider 2-Calabi-Yau categories with a directed cluster-tilting subcategory; we study their cluster-tilting subcategories and the cluster combinatorics that they encode. We show that such categories have a cluster structure. Triangulated 2-Calabi-Yau categories with a directed cluster-tilting subcategory are closely related to representations of certain semi-hereditary categories, more specifically to representations of thread quivers. Thread quivers are a tool to classify and study certain semi-hereditary categories using both quivers and linearly ordered sets (threads). We study the case where the thread quiver consists of a single thread (so that representations of this thread quiver correspond to representations of some linearly ordered set), and show that, similar to the case of a Dynkin quiver of type $A$, the cluster-tilting subcategories can be understood via triangulations of an associated cyclically ordered set. In this way, we gain insight into the structure of the cluster-tilting subcategories of 2-Calabi-Yau categories with a directed cluster-tilting subcategory. As an application, we show that every 2-Calabi-Yau category which admits a directed cluster-tilting subcategory with countably many isomorphism classes of indecomposable objects has a cluster-tilting subcategory $\mathcal{V}$ with the following property: any rigid object in the cluster category can be reached from $\mathcal{V}$ by finitely many mutations. This implies that there is a cluster map which is defined on all rigid objects, and thus that there is a cluster algebra whose cluster variables are exactly given by the rigid indecomposable objects.