Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In the preceding part (I) of this paper,  shod that for any torsion pair (i.e., t-structure without the shift-closedness) in a triangulated category, there is an associated abelian category, which  call the heart. Two extremal cases of torsion pairs are t-structures and cluster tilting subcategories. If the torsion pair comes from a t-structure, then its heart is nothing other than the heart of this t-structure. In this case, as is ll known, by composing certain adjoint functors,  obtain a homological functor from the triangulated category to the heart. If the torsion pair comes from a cluster tilting subcategory, then its heart coincides with the quotient category of the triangulated category by this subcategory. In this case, the quotient functor becomes homological. ,  unify these two constructions, to obtain a homological functor from the triangulated category, to the heart of any torsion pair.
['Noriyuki Abe', 'Hiroyuki Nakaoka']
I am writing a research paper on a idea General Heart Construction on a Triangulated Category (II): Associated Homological Functor
[{"from": "human", "value": "I am writing a research paper on a idea General Heart Construction on a Triangulated Category (II): Associated Homological Functor"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In the preceding part (I) of this paper,  shod that for any torsion pair (i.e., t-structure without the shift-closedness) in a triangulated category, there is an associated abelian category, which  call the heart. Two extremal cases of torsion pairs are t-structures and cluster tilting subcategories. If the torsion pair comes from a t-structure, then its heart is nothing other than the heart of this t-structure. In this case, as is ll known, by composing certain adjoint functors,  obtain a homological functor from the triangulated category to the heart. If the torsion pair comes from a cluster tilting subcategory, then its heart coincides with the quotient category of the triangulated category by this subcategory. In this case, the quotient functor becomes homological. ,  unify these two constructions, to obtain a homological functor from the triangulated category, to the heart of any torsion pair."}]