Postnikov towers, k-invariants and obstruction theory for dg categories

By inspiring ourselves in Drinfeld's DG quotient, we develop Postnikov towers, k-invariants and an obstruction theory for dg categories. As an application, we obtain the following ‘rigidification’ theorem: let A be a homologically connective dg category and a dg functor to its homotopy category. If the inductive family {ωn(Fn)}n⩾0 of obstruction classes vanishes, then a lift for F0 exists.