Geodesic
Totality of colimit closures
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 4, pp. 761-768.

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Adámek, Herrlich, and Reiterman showed that a cocomplete category $\Cal A$ is cocomplete if there exists a small (full) subcategory $\Cal B$ such that every $\Cal A$-object is a colimit of $\Cal B$-objects. The authors of the present paper strengthened the result to totality in the sense of Street and Walters. Here we weaken the hypothesis, assuming only that the colimit closure is attained by transfinite iteration of the colimit closure process up to a fixed ordinal. This requires some investigations on generalized notions of generators.
Classification : 18A20, 18A30, 18A35, 18A40, 18B99
Keywords: cocomplete category; (almost-)$\Cal E$-generator; colimit closure; cointersection; total category 
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Börger, Reinhard; Tholen, Walter. Totality of colimit closures. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 4, pp. 761-768. http://geodesic.mathdoc.fr/item/CMUC_1991__32_4_a18/