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Notes on enriched categories with colimits of some class. (English) Zbl 1082.18004

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.

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)
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