Reductions of triangulated categories and simple‐minded collections

Silting and Calabi–Yau reductions are important processes in representation theory to construct new triangulated categories from given ones, which similar Verdier quotient. In this paper, first we introduce a reduction process of category, is analogous the silting (Calabi–Yau) reduction. For category T $\mathcal {T}$ with pre-simple-minded collection (pre-SMC) R {R}$ , U {U}$ such that SMCs bijectively correspond those containing . Second, give an analogue Buchweitz's theorem for singularity sg {T}_{\operatorname{sg}\nolimits }$ SMC quadruple ( p S ) $(\mathcal {T},\mathcal {T}^{\mathrm{p}},\mathbb {S}, \mathcal {S})$ : can be realized as stable extriangulated subcategory F {F}$ Finally, show simple-minded system (SMS) due Coelho Simões Pauksztello shadow our This parallel result Iyama Yang.