Completeness of cocompletions
We know well that the existence of (finite) limits in a free cocompletion of a category \(\mathcal C\) under (finite) colimits is not automatically guaranteed. The existence of such limits is a kind of a weak completeness condition on the category \(\mathcal C\). The authors propose several weak completeness concepts. Then they show that all these concepts are equivalent in the sense that if the cone functors of all finite diagrams in a category satisfy simultaneously one of the proposed finiteness conditions, then they simultaneously satisfy any of them. The established equivalence of limit concepts is used so as to derive results on the existence of limits in colimit cocompletions. It turns out that there is a crucial difference according as we consider finite diagrams or \(\alpha\)-small ones for uncountable \(\alpha\).
scientific article; zbMATH DE number 2153394
scientific article