Continuously triangulating the continuous cluster category

TL;DR

This paper classifies all possible continuous triangulations of finite coverings of the continuous cluster category, revealing new structures and covering spaces that extend the understanding of its topological and categorical properties.

Contribution

It provides a complete classification of continuous triangulations for finite coverings of the continuous cluster category, including new and unexpected coverings.

Findings

Classified all continuous triangulations of finite coverings.

Identified a new add-triangulated 2-fold covering of the Moebius strip category.

Extended the understanding of topological structures in continuous cluster categories.

Abstract

In [4], the continuous cluster category was introduced. This is a topological category whose space of isomorphism classes of indecomposable objects forms a Moebius band. It was found in [4] that, in order to have a continuously triangulated structure on this category, one needs at least two copies of each indecomposable object forming a 2-fold covering space of the Moebius band. This paper classifies all continuous triangulations of finite coverings of the basic continuous cluster category. This includes the connected 2-fold covering of Igusa-Todorov [4], the disconnected 2-fold covering of Orlov [6] and a third unexpected continuously add-triangulated 2-fold covering of the Moebius strip category.