Negative cluster categories from simple minded collection quadruples

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.