The closure of a class of colimits
The authors consider \({\mathcal V}\)-categories, where \({\mathcal V}\) is a symmetric monoidal closed category. Write \(\phi\) *T for the colimit of T: \({\mathcal K}\to {\mathcal A}\) indexed by \(\phi\) : \({\mathcal K}^{op}\to {\mathcal V}\). Suppose that \(\Phi\) is a class of such indexing types (\({\mathcal K},\phi)\) and write \(\Phi^*\) for the class of indexing types (\({\mathcal F},\psi)\) such that every \(\Phi\)-cocomplete \({\mathcal A}\) is \(\psi\)-cocomplete and every \(\Phi\)-cocontinuous functor is \(\psi\)-cocontinuous. The authors show that \(\psi\in [{\mathcal F}^{op},{\mathcal V}]\) lies in \(\Phi^*\) if and only if it lies in the \(\Phi\)-colimit closure of \({\mathcal F}\) in [\({\mathcal F}^{op},{\mathcal V}]\), and they characterize those \(\Phi\) for which \(\Phi^*=\Phi\). They also comment on a similar sort of questions and pose open problems.
scientific article; zbMATH DE number 4071156