Continuously triangulating the continuous cluster category
arXiv:1907.11365
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.
49 pages, 3 figures, earlier versions of this work were presented by the first author at the Auslander Conference at Woods Hole in May, 2015 and by the second author at a workshop at Tsinghua University in Beijing in July 2017. Version 1 was part of the first author's PhD thesis. Submitted to Topology and its Applications