Realization functors in algebraic triangulated categories

Abstract

Let \({\mathcal {T}}\) be an algebraic triangulated category and \({\mathcal {C}}\) an extension-closed subcategory with \({{\,\textrm{Hom}\,}}({\mathcal {C}}, \Sigma ^{<0} {\mathcal {C}})=0\). Then \({\mathcal {C}}\) has an exact structure induced from exact triangles in \({\mathcal {T}}\). Keller and Vossieck say that there exists a triangle functor \(\operatorname {D}^{b}({\mathcal {C}}) \rightarrow {\mathcal {T}}\) extending the inclusion \({\mathcal {C}} \subseteq {\mathcal {T}}\). We provide the missing details for a complete proof.