Negative cluster categories from simple minded collection quadruples

TL;DR

This paper explores the structure of quotient categories derived from simple minded collection quadruples, revealing their Hom-space relationships and establishing conditions under which they form negative cluster categories.

Contribution

It introduces a new approach using limits and colimits to analyze Hom-spaces in quotient categories from simple minded collection quadruples, extending the understanding of negative-Calabi-Yau triples.

Findings

Hom-spaces over $ /T^p$ relate to those over $ $ via limits and colimits

$ /T^p$ is a negative cluster category when derived from a negative-Calabi-Yau triple

Provides a new proof for Jin's result on negative-Calabi-Yau triples

Abstract

Fomin and Zelevinsky's definition of cluster algebras laid the foundation for cluster theory. The various categorifications and generalisations of the original definition led to Iyama and Yoshino's generalised cluster categories $\mathcal{T}/\mathcal{T}^{fd}$ coming from positive-Calabi-Yau triples $(\mathcal{T}, \mathcal{T}^{fd},\mathcal{M})$. Jin later defined simple minded collection quadruples $(\mathcal{T}, \mathcal{T}^{p},\mathbb{S},\mathcal{S})$, where the special case $\mathbb{S}=\Sigma^{-d}$ is the analogue of Iyama and Yang's triples: negative-Calabi-Yau triples. In this paper, we further study the quotient categories $\mathcal{T}/\mathcal{T}^p$ coming from simple minded collection quadruples. Our main result uses limits and colimits to describe Hom-spaces over $\mathcal{T}/\mathcal{T}^p$ in relation to the easier to understand Hom-spaces over $\mathcal{T}$. Moreover, we apply our theorem to give a different proof of a result by Jin: if we have a negative-Calabi-Yau triple, then $\mathcal{T}/\mathcal{T}^p$ is a negative cluster category.