Abelian Categories from Triangulated Categories via Nakaoka–Palu’s Localization

The aim of this paper is to provide an expansion of Abe–Nakaoka’s heart construction to the following two different realizations of the module category over the endomorphism ring of a rigid object in a triangulated category: Buan–Marsh’s localization and Iyama–Yoshino’s subfactor. Our method depends on a modification of Nakaoka–Palu’s HTCP localization, a Gabriel–Zisman localization of extriangulated categories which is also realized as a subfactor of the original ones. Besides of the heart construction, our generalized HTCP localization involves the following phenomena: (1) stable category with respect to a class of objects; (2) recollement of triangulated categories; (3) recollement of abelian categories under a certain assumption.