Negative cluster categories from simple minded collection quadruples

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