# Cyclic posets and triangulation clusters

## TL;DR

This paper explores the structure of triangulation clusters derived from cyclic posets, connecting algebraic and topological triangulations, including those of cactus spaces, to advance understanding of cluster categories.

## Contribution

It provides a comprehensive analysis of triangulation clusters from cyclic posets, extending to topological triangulations of cactus spaces, and clarifies their algebraic and topological relationships.

## Key findings

- Triangulation clusters correspond to topological triangulations of the 2-disk.
- Locally finite non-triangulation clusters relate to cactus space triangulations.
- The paper generalizes cluster structures in triangulated categories from cyclic posets.

## Abstract

Triangulated categories coming from cyclic posets were originally introduced by the authors in [IT15b] as a generalization of the constructions of various triangulated categories with cluster structures. We give an overview, then analyze triangulation clusters which are those corresponding to topological triangulations of the 2-disk. Locally finite non-triangulation clusters give topological triangulations of the cactus space associated to the cactus cyclic poset.