Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Cartesian closed topological hulls


Authors: H. Herrlich and L. D. Nel
Journal: Proc. Amer. Math. Soc. 62 (1977), 215-222
MSC: Primary 18D15
MathSciNet review: 0476831
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown in this paper that if a concrete category $ \mathfrak{A}$ admits embedding as a full finitely productive subcategory of a cartesian closed topological (CCT) category, then $ \mathfrak{A}$ admits such embedding into a smallest CCT category, its CCT hull. This hull is characterized internally by means of density properties and externally by means of a universal property. The problem is posed of whether every topological category has a CCT hull.


References [Enhancements On Off] (What's this?)

  • [1] P. Antoine, Étude élémentaire des catégories d'ensembles structurés. I, II, Bull. Soc. Math. Belge 18 (1966), 142-164; ibid 18 (1966), 387-414. MR 34 #220; 35 #4135.
  • [2] Andrée Bastiani, Applications différentiables et variétés différentiables de dimension infinie, J. Analyse Math. 13 (1964), 1–114 (French). MR 0177277
  • [3] H. L. Bentley, H. Herrlich, and W. A. Robertson, Convenient categories for topologists, Comment. Math. Univ. Carolinae 17 (1976), no. 2, 207–227. MR 0425890
  • [4] E. Binz, Bemerkungen zu limitierten Funktionenalgebren, Math. Ann. 175 (1968), 169–184 (German). MR 0221461
  • [5] -, Continuous convergence on $ C(X)$, Lecture Notes in Math., vol. 469, Springer-Verlag, Berlin and New York, 1975.
  • [6] E. Binz and H. H. Keller, Funktionenräume in der Kategorie der Limesräume, Ann. Acad. Sci. Fenn. Ser. A I No. 383 (1966), 21 (German). MR 0206890
  • [7] Gérard Bourdaud, Espaces d’Antoine et semi-espaces d’Antoine, Cahiers Topologie Géom. Différentielle 16 (1975), no. 2, 107–133 (French). MR 0394529
  • [8] Gérard Bourdaud, Deux caractérisations des 𝑐-espaces, C. R. Acad. Sci. Paris Sér. A-B 281 (1975), no. 10, Ai, A313–A316 (French, with English summary). MR 0380700
  • [9] Gérard Bourdaud, Some Cartesian closed topological categories of convergence spaces, Categorical topology (Proc. Conf., Mannheim, 1975) Springer, Berlin, 1976, pp. 93–108. Lecture Notes in Math., Vol. 540. MR 0493924
  • [10] Mario Chartrelle, Construction de catégories auto-dominées, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A388–A391 (French). MR 0338118
  • [11] G. Choquet, Convergences, Ann. Univ. Grenoble. Sect. Sci. Math. Phys. (N.S.) 23 (1948), 57–112. MR 0025716
  • [12] C. H. Cook and H. R. Fischer, On equicontinuity and continuous convergence, Math. Ann. 159 (1965), 94–104. MR 0179752
  • [13] Brian Day, A reflection theorem for closed categories, J. Pure Appl. Algebra 2 (1972), no. 1, 1–11. MR 0296126
  • [14] H. R. Fischer, Limesräume, Math. Ann. 137 (1959), 269–303 (German). MR 0109339
  • [15] Horst Herrlich and George E. Strecker, Category theory: an introduction, Allyn and Bacon Inc., Boston, Mass., 1973. Allyn and Bacon Series in Advanced Mathematics. MR 0349791
  • [16] Horst Herrlich, Topological functors, General Topology and Appl. 4 (1974), 125–142. MR 0343226
  • [17] Horst Herrlich, Cartesian closed topological categories, Math. Colloq. Univ. Cape Town 9 (1974), 1–16. MR 0460414
  • [18] Horst Herrlich, Some topological theorems which fail to be true, 1976. MR 0436073
  • [19] Horst Herrlich, Initial completions, Math. Z. 150 (1976), no. 2, 101–110. MR 0437614
  • [20] M. Katětov, On continuity structures and spaces of mappings, Comment. Math. Univ. Carolinae 6 (1965), 257–278. MR 0193608
  • [21] D. Kent, Convergence functions and their related topologies, Fund. Math. 54 (1964), 125–133. MR 0161301
  • [22] D. Kent, K. McKennon, G. Richardson, and M. Schroder, Continuous convergence in 𝐶(𝑋), Pacific J. Math. 52 (1974), 457–465. MR 0370482
  • [25] Hans-Joachim Kowalsky, Limesräume und Komplettierung, Math. Nachr. 12 (1954), 301–340 (German). MR 0073147
  • [24] Armando Machado, Espaces d’Antoine et pseudo-topologies, Cahiers Topologie Géom. Différentielle 14 (1973), 309–327 (French). MR 0345054
  • [25] D. Marxen, Quotients of quasi-uniform spaces (preprint).
  • [26] H. Müller, Über die Vertauschbarkeit von Reflexionen und Coreflexionen Quasi-Top Kategorien über Meng (preprint).
  • [27] L. D. Nel, Initially structured categories and Cartesian closedness, Canad. J. Math. 27 (1975), no. 6, 1361–1377. MR 0393183
  • [28] L. D. Nel, Cartesian closed topological categories, Categorical topology (Proc. Conf., Mannheim, 1975) Springer, Berlin, 1976, pp. 439–451. Lecture Notes in Math., Vol. 540. MR 0447369
  • [29] H. Poppe, Compactness in general function spaces, VEB Deutscher Verlag der Wissenschaften, Berlin, 1974. MR 0438281
  • [30] M. Schroder, Solid convergence spaces, Bull. Austral. Math. Soc. 8 (1973), 443–459. MR 0320995
  • [31] E. Spanier, Quasi-topologies, Duke Math. J. 30 (1963), 1–14. MR 0144300
  • [32] Oswald Wyler, Convenient categories for topology, General Topology and Appl. 3 (1973), 225–242. MR 0324622
  • [33] Oswald Wyler, Are there topoi in topology?, Categorical topology (Proc. Conf., Mannheim, 1975) Springer, Berlin, 1976, pp. 699–719. Lecture Notes in Math., Vol. 540. MR 0458346

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 18D15

Retrieve articles in all journals with MSC: 18D15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0476831-6
Keywords: Cartesian closed topological hull, dense functorial embedding, initial source preserving, power preserving, concrete category
Article copyright: © Copyright 1977 American Mathematical Society