Abstract
Triangulated categories coming from cyclic posets were originally introduced by the authors in a previous paper as a generalization of the constructions of various triangulated categories with cluster structures. We give an overview, and then analyze “triangulation clusters” which are those corresponding to topological triangulations of the 2-disk. Locally finite nontriangulation clusters give topological triangulations of the “cactus space” associated to the “cactus cyclic poset”.