On Amiot's conjecture

TL;DR

This paper proves a variant of Amiot's conjecture, providing a categorical characterization of cluster categories and their higher-dimensional and relative variants, thus advancing the understanding of cluster algebra categorification.

Contribution

It offers the first proof of Amiot's conjecture and extends it to higher-dimensional and relative Higgs categories, broadening the theoretical framework.

Findings

Proof of a variant of Amiot's conjecture for cluster categories

Extension of the conjecture to Higgs categories

Establishment of higher-dimensional and relative category variants

Abstract

In a survey paper in 2011, Amiot proposed a conjectural characterisation of the cluster categories which were conceived in the mid 2000s to lift the combinatorics of Fomin-Zelevinsky's cluster algebras to the categorical level. This paper is devoted to a proof of (a variant of) her conjecture. More generally, cluster categories admit higher-dimensional and relative variants, the so-called Higgs categories recently introduced by Wu. We also prove higher-dimensional and relative variants of the conjecture.