Progress on Infinite Cluster Categories Related to Triangulations of the (Punctured) Disk

TL;DR

This paper surveys recent developments in infinite cluster categories related to disk triangulations, introduces new infinite families of such categories, and connects combinatorial models with algebraic constructions.

Contribution

It introduces two new infinite families of weak cluster categories of type D, linking combinatorial models with algebraic representations in the infinite setting.

Findings

Combinatorics of cluster categories match punctured disk triangulations

Construction of new infinite cluster categories from thread quivers

Formulation of conjectures on weak cluster structures

Abstract

In this mostly expository paper, we present recent progress on infinite (weak) cluster categories that are related to triangulations of the disk, with and without a puncture. First we recall the notion of a cluster category. Then we move to the infinite setting and survey recent work on infinite cluster categories of types $\mathbb{A}$ and $\mathbb{D}$. We conclude with our contributions, two infinite families of infinite (weak) cluster categories of type $\mathbb{D}$. We first present a discrete, infinite version of Schiffler's combinatorial model of the punctured disk with marked points. We then produce each (weak) cluster category starting with representations of thread quivers, taking the derived category, and then taking the appropriate orbit category. We show that the combinatorics in the (weak) cluster categories match with the corresponding combinatorics of the punctured disk with countably-many marked points. We also state two conjectures concerning weak cluster structures inside our (weak) cluster categories.