Kelly, G. M.; Schmitt, V. Notes on enriched categories with colimits of some class. (English) Zbl 1082.18004 Theory Appl. Categ. 14, 399-423 (2005). While research for this paper began as an analysis of the F. Borceux, C. Quinteiro and J. Rosický article [“A theory of enriched sketches”, Theory Appl. Categ. 4, 47–72 (1998; Zbl 0981.18006)], it has become a self-contained survey of (enriched) categories admitting colimits weighted by members of some class \(\Phi\), and contains quite a few new results. In particular, a weight is called \(\Phi\)-flat when its colimits in the base \(\mathcal{V}\) commute with \(\Phi\)-weighted limits, while a weight is called \(\Phi\)-atomic when its limits in \(\mathcal{V}\) commute with \(\Phi\)-weighted colimits. A Morita theorem is proved for any class \(\Phi\) containing the absolute weights [R. Street, “Absolute colimits in enriched categories”, Cah. Topol. Géom. Différ. 24, 377–379 (1983; Zbl 0532.18001)]. The relationship between \(\Phi\)-continuous and \(\Phi\)-flat weights is studied. Reviewer: Ross H. Street (North Ryde) Cited in 2 ReviewsCited in 44 Documents MSC: 18A35 Categories admitting limits (complete categories), functors preserving limits, completions 18C35 Accessible and locally presentable categories 18D20 Enriched categories (over closed or monoidal categories) Keywords:weights; colimits; limits; enriched category; Cauchy completion; flat; atomic; small presentable Citations:Zbl 0981.18006; Zbl 0532.18001 PDFBibTeX XMLCite \textit{G. M. Kelly} and \textit{V. Schmitt}, Theory Appl. Categ. 14, 399--423 (2005; Zbl 1082.18004) Full Text: arXiv EuDML EMIS