Localization of triangulated categories with respect to extension-closed subcategories

The aim of this paper is to develop a framework for localization theory of triangulated categories \(\mathcal{C}\), that is, from a given extension-closed subcategory \(\mathcal{N}\) of \(\mathcal{C}\), we construct a natural extriangulated structure on \(\mathcal{C}\) together with an exact functor \(Q:\mathcal{C}\to\widetilde{\mathcal{C}}_\mathcal{N}\) satisfying a suitable universality, which unifies several phenomena. Precisely, a given subcategory \(\mathcal{N}\) is thick if and only if the localization \(\widetilde{\mathcal{C}}_\mathcal{N}\) corresponds to a triangulated category. In this case, \(Q\) is nothing other than the usual Verdier quotient. Furthermore, it is revealed that \(\widetilde{\mathcal{C}}_\mathcal{N}\) is an exact category if and only if \(\mathcal{N}\) satisfies a generating condition \(\mathsf{cone}(\mathcal{N},\mathcal{N})=\mathcal{C}\). Such an (abelian) exact localization \(\widetilde{\mathcal{C}}_\mathcal{N}\) provides a good understanding of some cohomological functors \(\mathcal{C}\to\mathsf{Ab}\), e.g., the heart of \(t\)-structures on \(\mathcal{C}\) and the abelian quotient of \(\mathcal{C}\) by a cluster-tilting subcategory \(\mathcal{N}\).