Geodesic
Totality of colimit closures
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 4, pp. 761-768 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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/

[1] Adámek J., Herrlich H., Reiterman J.: Cocompleteness almost implies completeness. Proc. Conf. Cat. Top. Prague, World Scientific, Singapore, 1989. | MR

[2] Börger R.: Making factorizations compositive. Comment. Math. Univ. Carolinae 32 (1991), 749-759. | MR

[3] Börger R., Tholen W.: Concordant-dissonant and monotone-light. Proceedings of the International Conference on Categorical Topology, Toledo (Ohio), 1983, Sigma Series in Pure Mathematics 5 (1984), 90-107. | MR

[4] Börger R., Tholen W.: Total categories and solid functors. Canad. J. Math. 42 (1990), 213-229. | MR

[5] Börger R., Tholen W.: Strong, regular, and dense generators. Cahiers Topologie Géom. Différentielle Catégoriques, to appear. | MR

[6] Day B.: Further criteria for totality. Cahiers Topologie Géom. Différentielle Catégoriques 28 (1987), 77-78. | MR | Zbl

[7] Isbell J.R.: Structure of categories. Bull. Amer. Math. Soc. 72 (1966), 619-655. | MR | Zbl

[8] Kelly G.M.: Monomorphisms, epimorphisms, and pullbacks. J. Austral. Math. Soc. A9 (1969), 124-142. | MR

[9] Kelly G.M.: A survey on totality for enriched and ordinary categories. Cahiers Topologie Géom. Différentielle Catégoriques 27 (1986), 109-131. | MR

[10] Kunen K.: Set theory. Studies in Logic and the Foundation of Mathematics 102, North-Holland, Amsterdam, 1980. | MR | Zbl

[11] MacDonald J., Stone A.: Essentially monadic adjunctions. Lecture Notes in Mathematics 962, Springer, Berlin (1982), 167-174. | MR | Zbl

[12] Pareigis B.: Categories and Functors. Academic Press, London, 1970. | MR | Zbl

[13] Street R., Walters R.: Yoneda structures on 2-categories. J. Algebra 50 (1978), 350-379. | MR | Zbl