The closure of a class of colimits
type: Item
instance of: scholarly article
title: The closure of a class of colimits
zbMATH Open document ID: 0656.18004
DOI: 10.1016/0022-4049(88)90073-4
author: Max Kelly
author: Michael Albert
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 18D15
Mathematics Subject Classification ID: 18A30
zbMATH DE Number: 4071156
zbMATH Keywords: symmetric monoidal closed category
zbMATH Keywords: cocontinuous functor
reviewed by: Pavel Pták
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90073-4
OpenAlex ID: W1993559687
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description: scientific article; zbMATH DE number 4071156