On preaccessible categories
For a regular cardinal \(\lambda\), a category is \(\lambda\)-preaccessible if it has a small set \(G\) of \(\lambda\)-presentable objects such that every object is a \(\lambda\)-directed colimit of objects in \(G\); hence this notion drops the required existence of all \(\lambda\)-directed colimits from the familiar notion of \(\lambda\)-accessibility. An easy example of a preaccessible but not accessible category is provided by any small category in which idempotents do not split. A preaccessible category has a dense generating set, and the converse proposition holds true if (and only if) the set-theoretic Vopěnka principle holds true. This principle makes every preaccessible category with equalizers accessible and therefore axiomatizable [in the sense of \textit{M. Makkai} and \textit{R. Paré}, ``Accessible categories: The foundations of categorical model theory'', Contemp. Math. 104 (1989; Zbl 0703.03042)], but it is not known whether every axiomatizable category is preaccessible.
scientific article; zbMATH DE number 858618