Let be a differential graded category.
A twisted complex in is
a graded set of objects of , such that only finitely many are not the zero object;
a set of morphisms such that
The differential graded category of twisted complexes in has as objects twisted complexes and
with differential given on given by
The construction of categories of twisted complexes is functorial in that for a dg-functor, there is a dg-functor
Passing from a dg-category to its category of twisted complexes is a step towards enhancing it to a pretriangulated dg-category.