T-structures on unbounded twisted complexes

This paper is a sequel to T-structures and twisted complexes on derived injectives by the same author with W. Lowen and M. Van den Bergh. We define a dg-category of unbounded twisted complexes on a dg-category, which is particularly interesting in the case of dg-categories of derived injectives or derived projectives associated to a t-structure. On such unbounded twisted complexes we define a natural “injective” and dually a “projective” t-structure. This is intended as a direct generalization of the homotopy categories of injective or projective objects of an abelian category.