T-structures on dg-categories and derived deformations

This paper is a sequel to "t-structures and twisted complexes on derived injectives" by the same authors. We develop the foundations of the infinitesimal derived deformation theory of pretriangulated dg-categories endowed with t-structures. This generalizes the deformation theory of abelian categories developed by the last two authors. We show how deformations of dg-categories of derived injectives yield derived deformations of the associated t-structures.