Tropical friezes and the index in higher homological algebra

Abstract: 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 $d \geqslant 1$. 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.