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.

中文翻译：

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连续对连续聚类类别进行三角剖分

在[4]中，引入了连续聚类类别。这是一个拓扑范畴，其不可分解对象的同构类空间形成莫比乌斯带。在[4]中发现，为了在该类别上具有连续的三角结构，每个不可分解物体至少需要两个副本，形成莫比乌斯带的2倍覆盖空间。本文对基本连续簇范畴的有限覆盖的所有连续三角剖分进行了分类。这包括 Igusa-Todorov [4] 的连接的 2-fold 覆盖、Orlov [6] 的断开的 2-fold 覆盖和 Moebius 带类别的第三个意想不到的连续添加三角化的 2-fold 覆盖。