Cauchy completeness for DG-categories

Abstract

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.

27 pages. Remark 7.3 is new. It gives an example to show that cokernels of protosplit chain maps cannot be constructed from the other assumed colimits appearing in Theorem 7.2. Thanks to the referee for pointing out other needed changes