  We go back to the roots of enriched category theory and study categories
enriched in chain complexes; that is, we deal with differential graded
categories (DG-categories for short). In particular, we recall weighted
colimits and provide examples. We solve the 50 year old question of how to
characterize Cauchy complete DG-categories in terms of existence of some
specific finite absolute colimits. As well as the interactions between absolute
weighted colimits, we also examine the total complex of a chain complex in a
DG-category as a non-absolute weighted colimit.
