Tilting objects in triangulated categories

Based on Beligiannis’s theory in [Beligiannis, A. (2000). Relative homological algebra and purity in triangulated categories. J. Algebra 227(1):268–361], we introduce and study E-tilting objects in a triangulated category, where E is a proper class of triangles. We show that each E-tilting object cogenerates an E-cotorsion pair. Meanwhile, we also achieve some nice characterizations with respect to the E-tilting object. As an application, we provide a necessary and sufficient condition for a triangulated category to be E-1-Gorenstein. Finally, we give a one to one correspondence between the class of E-tilting objects and the class of tilting subcategories in a suitable functor category.