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.
Keywords: chain complex; suspension; mapping cone; differential graded algebra; Cauchy completion; DG-category; absolute colimit; total complex
2020 MSC: 18D20; 18G35; 18E30; 55P42
Theory and Applications of Categories, Vol. 37, 2021, No. 28, pp 940-963.