Monoidal Morita equivalence (Q584383)

Morita equivalence has been studied for categories enriched over a monoidal category. The author considers enriched categories possessing themselves a monoidal structure and derive many of the Morita theorems in this context. The key used is the definition of a monoidal Cauchy completion. Im and Kelly have studied the free monoidal cocompletion of a small enriched category. Much of their work extends to free monoidal \b{F}-cocompletions where \b{F} is any set of weights for colimits. One has also to use an observation of R. Street that the Cauchy completion is just the free cocompletion under absolute colimits, to enrich the monoidal structure on the Cauchy completion of any small monoidal category. Finally, conditions under which the monoidal Cauchy completion is closed are also discussed.