Small Cauchy completions (Q910488)

Cauchy completions in the categorical sense first appeared in \textit{F. W. Lawvere}'s study of generalized metric spaces, i.e. categories enriched over \(\bar R_+\) [Rend. Semin. Mat. Fis. Milano 43, 135-166 (1973; Zbl 0335.18006)]. They have since become an important concept in the general theory of categories enriched over a ``nice'' base category \({\mathcal V}\), where ``nice'' usually means symmetric monoidal and/or closed as well as complete and cocomplete. For \({\mathcal V}=Set\), \({\mathcal V}=Ab\) and \({\mathcal V}=\bar R_+\) the Cauchy completions of small \({\mathcal V}\)-categories are again small, but if \({\mathcal V}\) is the category of complete join semilattices equipped with a suitable tensor product, this need not be the case [cf. \textit{G. M. Kelly}, Basic concepts of enriched category theory (1982; Zbl 0478.18005)]. Now the question arises, under what conditions smallness is preserved under the operation of Cauchy completion. The author improves upon unpublished results by Kelly and gives an affirmative answer in the case that \({\mathcal V}\) is locally presentable.