Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander–Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with marked points.Cluster characters are maps from cluster categories (and more general triangulated categories) to cluster algebras. They have a tropical shadow in the form of so-called tropical friezes, which are maps from cluster categories (and more general triangulated categories) to the integers.This paper will define higher dimensional tropical friezes. One of the motivations is the higher dimensional cluster categories of Oppermann and Thomas, which encode (d + 1)-dimensional structures for an integer d ⩾ 1. They are (d + 2)-angulated categories, which belong to the subject of higher homological algebra.We will define higher dimensional tropical friezes as maps from higher cluster categories (and more general (d + 2)-angulated categories) to the integers. Following Palu, we will define a notion of (d + 2)-angulated index, establish some of its properties, and use it to construct higher dimensional tropical friezes.

中文翻译：

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热带带状饰带与高等同调代数中的指数

簇类别和簇代数编码二维结构。例如，可以在曲面上绘制聚类类别的 Auslander-Reiten 箭袋，并且存在由带有标记点的曲面确定的一类聚类代数。聚类特征是从聚类类别（以及更一般的三角剖分类别）到聚类的映射代数。它们具有所谓的热带带状热带阴影，这是从簇类别（以及更一般的三角分类）到整数的映射。本文将定义更高维的热带带状带。动机之一是 Oppermann 和 Thomas 的高维集群类别，它们编码 (d+ 1) 整数的维结构d⩾ 1. 它们是 (d+ 2)-有角度的类别，属于更高同调代数的主题。我们将更高维的热带带状饰带定义为来自更高集群类别的地图（更一般的（d+ 2）-角度类别）到整数。在 Palu 之后，我们将定义一个 (d+ 2)-角度索引，建立它的一些属性，并用它来构建更高维度的热带带状饰带。