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Total Categories and Solid Functors

Published online by Cambridge University Press:  20 November 2018

Reinhard Börger  and
Walter Tholen
Reinhard Börger
Affiliation:
Fernuniversität, Hagen, Federal Republic of Germany
Walter Tholen
Affiliation:
York UniversityNorth York, Ontario
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Totality of a category as introduced by Street and Walters [17] is known to be a strong cocompleteness property (cf. also [21]) which goes far beyond ordinary (small) cocompleteness. It implies compactness in the sense of Isbell [11] and therefore hypercompleteness [7], that is: the existence of limits of all those (not necessarily small) diagrams which are not prevented from having a limit merely from size-considerations with respect to the homsets. In particular, arbitrary intersections of monomorphisms exist in a total category; which is part of Street's [16] characterization of totality and is used in establishing the interrelationship with topoi (cf. also [15]).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 42 , Issue 2 , 01 April 1990 , pp. 213 - 229
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Adámek, J., Colimits of algebras revisited, Bull. Austral. Math. Soc. 17 (1977), 433–450.Google Scholar
2. Adámek, J., How complete are categories in algebras? Bull. Austral. Math. Soc. 36 (1987), 389-09.Google Scholar
3. Anghel, C., Factorizations and initiality in enriched categories, Doctoral dissertation, Fernuniversität Hagen (1987).Google Scholar
4. Anghel, C., Lifting properties of V-functors, preprint (1987).Google Scholar
5. Börger, R. and Tholen, W., Cantors Diagonalprinzip für Kategorien, Math. Z. 160 (1978), 135- 138.Google Scholar
6. Börger, R. and Tholen, W., Strong, regular, and dense generators, Cahiers Topologie Géom. Différentielle Catégoriques (submitted).Google Scholar
7. Börger, R., Tholen, W., Wischenwsky, M.B. and Wolff, H., Compact and hypercomplete categories, J. Pure Appl. Algebra 21 (1981), 120-144.Google Scholar
8. Day, B.J., Further criteria for totality, Cahiers Topologie Géom. Différentielle Catégoriques 28 (1987), 77–78.Google Scholar
9. Gabriel, R and Ulmer, F., Lokalpräsentierbare Kategorien, Lecture Notes in Math. 221 (Springer- Verlag, Berlin-Heidelberg-New York, 1971).Google Scholar
10. Hoffmann, R.-E., Topological functors admitting generalized Cauchy-completion, Lecture Notes in Math. 540 (Springer-Verlag, Berlin-Heidelberg-New York, 1976) 286–344.Google Scholar
11. Isbell, J.R., Small subcategories and completeness, Math. Systems Theory 2 (1968), 27–50.Google Scholar
12. Kelly, G.M., A survey of totality for enriched and ordinary categories, Cahiers Topologie Géom. Différentielle Catégoriques 27 (1986), 109–131.Google Scholar
13. Mac Lane, S., Categories for the working mathematician (Springer-Verlag, Berlin-Heidelberg- New York, 1971).Google Scholar
14. Pareigis, B., Categories and functors (Academic Press, New York-London, 1970).Google Scholar
15. Street, R., Notions of topos, Bull. Austral. Math. Soc. 23 (1981), 199–208.Google Scholar
16. Street, R., The family approach to total cocompleteness and toposes, Trans. Amer. Math. Soc. 284 (1984), 355–369.Google Scholar
17. Street, R. and Walters, R., Yoneda structures on 2-categories, J. Algebra 50 (1978), 350–379.Google Scholar
18. Tholen, W., Semi-topological functors I, J. Pure Appl. Algebra 15 (1979), 53–73.Google Scholar
19. Tholen, W., Note on total categories, Bull. Austral. Math. Soc. 21 (1980), 169–173.Google Scholar
20. Tholen, W., Factorizations, localizations and the orthogonal subcategory. Google Scholar
21. Wood, R.J., Some remarks on total categories, J. Algebra 75 (1982), 538–545.Google Scholar