I would like to understand the following setting: for an additive $B$ localize $K^b(B)$ by a set of $B$-morphisms (i.e. by a thick triangulated subcategory generated by some set of two-term complexes). Are there any well known examples for such a category? In particular, I would like to know whether in this localization could exist non-zero morphism between $X[2]$ and $Y$, where $X$ and $Y$ come from $B$ (i.e. 'two-extensions of objects of $B$').

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