Totality of colimit closures

Reinhard Börger; Walter Tholen

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 4, page 761-768
  • ISSN: 0010-2628

Abstract

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Adámek, Herrlich, and Reiterman showed that a cocomplete category 𝒜 is cocomplete if there exists a small (full) subcategory ℬ such that every 𝒜 -object is a colimit of 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.

How to cite

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Börger, Reinhard, and Tholen, Walter. "Totality of colimit closures." Commentationes Mathematicae Universitatis Carolinae 32.4 (1991): 761-768. <http://eudml.org/doc/247313>.

@article{Börger1991,
abstract = {Adámek, Herrlich, and Reiterman showed that a cocomplete category $\mathcal \{A\}$ is cocomplete if there exists a small (full) subcategory $\mathcal \{B\}$ such that every $\mathcal \{A\}$-object is a colimit of $\mathcal \{B\}$-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.},
author = {Börger, Reinhard, Tholen, Walter},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {cocomplete category; (almost-)$\mathcal \{E\}$-generator; colimit closure; cointersection; total category; -generator; cointersection; total category; cocomplete category; colimit closure},
language = {eng},
number = {4},
pages = {761-768},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Totality of colimit closures},
url = {http://eudml.org/doc/247313},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Börger, Reinhard
AU - Tholen, Walter
TI - Totality of colimit closures
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 4
SP - 761
EP - 768
AB - Adámek, Herrlich, and Reiterman showed that a cocomplete category $\mathcal {A}$ is cocomplete if there exists a small (full) subcategory $\mathcal {B}$ such that every $\mathcal {A}$-object is a colimit of $\mathcal {B}$-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.
LA - eng
KW - cocomplete category; (almost-)$\mathcal {E}$-generator; colimit closure; cointersection; total category; -generator; cointersection; total category; cocomplete category; colimit closure
UR - http://eudml.org/doc/247313
ER -

References

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  1. Adámek J., Herrlich H., Reiterman J., Cocompleteness almost implies completeness. Proc. Conf. Cat. Top. Prague, World Scientific, Singapore, 1989, . MR1047905
  2. Börger R., Making factorizations compositive, Comment. Math. Univ. Carolinae 32 (1991), 749-759. (1991) MR1159822
  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. MR0785013
  4. Börger R., Tholen W., Total categories and solid functors, Canad. J. Math. 42 (1990), 213-229. (1990) MR1051726
  5. Börger R., Tholen W., Strong, regular, and dense generators, Cahiers Topologie Géom. Différentielle Catégoriques, to appear. MR1158111
  6. Day B., Further criteria for totality, Cahiers Topologie Géom. Différentielle Catégoriques 28 (1987), 77-78. (1987) Zbl0626.18001MR0903153
  7. Isbell J.R., Structure of categories, Bull. Amer. Math. Soc. 72 (1966), 619-655. (1966) Zbl0142.25401MR0206071
  8. Kelly G.M., Monomorphisms, epimorphisms, and pullbacks, J. Austral. Math. Soc. A9 (1969), 124-142. (1969) MR0240161
  9. Kelly G.M., A survey on totality for enriched and ordinary categories, Cahiers Topologie Géom. Différentielle Catégoriques 27 (1986), 109-131. (1986) MR0850527
  10. Kunen K., Set theory, Studies in Logic and the Foundation of Mathematics 102, North-Holland, Amsterdam, 1980. Zbl0960.03033MR0597342
  11. MacDonald J., Stone A., Essentially monadic adjunctions, Lecture Notes in Mathematics 962, Springer, Berlin (1982), 167-174. Zbl0498.18003MR0682954
  12. Pareigis B., Categories and Functors, Academic Press, London, 1970. Zbl0211.32402MR0265428
  13. Street R., Walters R., Yoneda structures on 2-categories, J. Algebra 50 (1978), 350-379. (1978) Zbl0401.18004MR0463261

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