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 are similar to Verdier quotient. In this paper, first we introduce a new reduction process of triangulated category, which is analogous to the silting (Calabi–Yau) reduction. For a triangulated category (Formula presented.) with a pre-simple-minded collection (pre-SMC) (Formula presented.), we construct a new triangulated category (Formula presented.) such that the SMCs in (Formula presented.) bijectively correspond to those in (Formula presented.) containing (Formula presented.). Second, we give an analogue of Buchweitz's theorem for the singularity category (Formula presented.) of a SMC quadruple (Formula presented.) : the category (Formula presented.) can be realized as the stable category of an extriangulated subcategory (Formula presented.) of (Formula presented.). Finally, we show the simple-minded system (SMS) reduction due to Coelho Simões and Pauksztello is the shadow of our SMC reduction. This is parallel to the result that Calabi–Yau reduction is the shadow of silting reduction due to Iyama and Yang.