  Silting and Calabi-Yau reductions are important process in representation
theory to construct new triangulated categories from given one, 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 $\cal T$ with a
pre-simple-minded collection (=pre-SMC) $\cal R$, we construct a new
triangulated category $\cal U$ such that the SMCs in $\cal U$ bijectively
correspond to those in $\cal T$ containing $\cal R$. Secondly, we give an
analogue of Buchweitz's theorem for the singularity category $\cal T_{\rm sg}$
of a SMC quadruple $(\cal T,\cal T^{\rm p},\mathbb S, \cal S)$: the category
$\cal T_{\rm sg}$ can be realized as the stable category of an extriangulated
subcategory $\cal F$ of $\cal T$. Finally, we show the SMS (simple-minded
system) reduction due to Coelho Sim\~oes 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.
