Metrics on triangulated categories and restrictions of (co)-$t$-structures

This paper explores the restriction behavior of silting-induced $t$-structures and co-$t$-structures on triangulated categories endowed with metrics. For compactly generated triangulated categories admitting small coproducts, silting subcategories of compact objects give rise to canonical $t$-structures. We establish that a silting subcategory being contravariantly finite in the precompletion (or completion) is equivalent to the canonical $t$-structure restricting to this precompletion (or completion). This result yields a purely categorical characterization of right coherent rings: a ring $R$ is right coherent if and only if the standard $t$-structure on $\mathcal{D}({\sf Mod}\text{-}R)$ restricts to a $t$-structure on $\mathcal{K}^{-,b}({\sf proj}\text{-}R)$. Furthermore, we show that the correspondences between silting objects, bounded (co)-$t$-structures, and simple-minded collections given by Koenig and Yang can be extended to the metric framework of triangulated categories, and still commute with mutation operations and preserve natural partial orders.