Cauchy completeness for DG-categories
Branko Nikoli\'c, Ross Street, Giacomo Tendas
TL;DR
This paper investigates Cauchy completeness in differential graded categories (DG-categories), characterizing it via finite absolute colimits and exploring the role of weighted colimits and total complexes.
Contribution
It provides a characterization of Cauchy complete DG-categories using finite absolute colimits, addressing a 50-year-old open problem in enriched category theory.
Findings
Characterization of Cauchy complete DG-categories via finite absolute colimits
Analysis of the interaction between absolute weighted colimits
Examination of total complexes as non-absolute weighted colimits
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.
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Taxonomy
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