Tilting objects in the extended heart of a t-structure

Building on the recent work of Adachi, Enomoto and Tsukamoto on a generalization of the Happel-Reiten-Smal{} tilting process, we study extended tilting objects in extriangulated categories with negative first extension. These objects coincide with the 1-tilting objects in abelian categories as in the work of Parra, Saor{\'i}n and Virili. We will be particularly interested in the case where the extriangulated category in question is the heart H_[t₁,t₂] of an interval of t-structures [t₁,t₂]. Our main results consist of a characterization of the extended tilting objects of a heart H_[t₁,t₂] for the case when t₂≤Σ⁻¹t₁, and another one for the case when Σ⁻²t₁<t₂. In the first one, we give conditions for these tilting objects to coincide with the quasi-tilting objects of the abelian category H_[t₁,Σ⁻¹t₁]. In the second one, it is given conditions for these to coincide with projective generators in the extriangulated category H_[t₁,₂]