t-structures and twisted complexes on derived injectives (Q2037603)

It is showed that an abelian category with enough injectives can be reconstructed as the category of finitely presented modules over the category of its injective objects [\textit{W. Lowen} et al., Trans. Am. Math. Soc. 358, No. 12, 5441--5483 (2006; Zbl 1113.13009)]. This result provides a natural path towards the deformation theory of abelian categories. The paper under review is the first one in an ongoing project to develop the deformation theory on triangulated categories with t-structures. The main result of the paper is a triangulated analogue of the result outlined above in the abelian case. Assume that \(\mathcal{T}\) is a triangulated category with a t-structure and with enough derived injectives, and moreover, \(\mathcal{T}=H^0(\mathcal{A})\) where \(\mathcal{A}\) is a pretriangulated dg category. The authors show that such dg category \(\mathcal{A}\) can be reconstructed in terms of so called ``bounded below twisted complexes'' on the full dg-subcategory of \(\mathcal{A}\) spanned by the derived injective objects. It is also proved that the ``reconstruction'' theorem can be enhanced to a functorial correspondence.