Entourages, Covers and Localic Groups

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• 作者：Jorge Picado (1)
  Ale? Pultr (2)
  
• 关键词：Frame ; Locale ; Frame coproduct ; Localic product ; Cover ; Entourage ; Cover and entourage uniformity ; Localic group ; 06D22 ; 18B30 ; 18B99 ; 18D35 ; 22A05 ; 54E15
• 刊名：Applied Categorical Structures
• 出版年：2013
• 出版时间：February 2013
• 年：2013
• 卷：21
• 期：1
• 页码：49-66
• 参考文献：1. Banaschewski, B., Hong, H.S., Pultr, A.: On the completion of nearness frames. Quaest. Math. 21, 19-7 (1998) CrossRef
  2. Banaschewski, B., Pultr, A.: Cauchy points of uniform and nearness frames. Quaest. Math. 19, 101-27 (1996) CrossRef
  3. Banaschewski, B., Pultr, A.: On Cauchy homomorphisms of nearness frames. Math. Nachr. 183, 5-8 (1997) CrossRef
  4. Banaschewski, B., Vermeulen, J.C.C.: On the completeness of localic groups. Comment. Math. Univ. Carol. 40, 293-07 (1999)
  5. Dowker, C.H., Strauss, D.: Sums in the category of frames. Houst. J. Math. 3, 7-5 (1977)
  6. Ferreira, M.J., Picado, J.: The Galois approach to uniform structures. Quaest. Math. 28, 355-73 (2005) CrossRef
  7. Fletcher, P., Hunsaker, W.: Entourage uniformities for frames. Monatsh. Math. 112, 271-79 (1991) CrossRef
  8. Fletcher, P., Hunsaker, W., Lindgren, W.: Characterizations of frame uniformities. Quaest. Math. 16, 371-83 (1993) CrossRef
  9. Frith, J.L.: Structured Frames. Ph.D. Thesis, University of Cape Town (1987)
  10. Isbell, J.R.: Uniform spaces. In: Mathematical Surveys, vol. 12. American Mathematical Society, New York (1964)
  11. Isbell, J.R.: Atomless parts of spaces. Math. Scand. 31, 5-2 (1972)
  12. Isbell, J.R., K?í?, I., Pultr, A., Rosicky, J.: Remarks on localic groups. In: Borceux, F. (ed.) Categorical Algebra and its Applications (Proc. Int. Conf. Louvain-La-Neuve), Lecture Notes in Math. 1348, pp. 154-72 (1987)
  13. Johnstone, P.T.: Stone Spaces, Cambridge Studies in Advanced Math. No. 3. Cambridge University Press, Cambridge (1983)
  14. Johnstone, P.T.: A simple proof that localic groups are closed. Cah. Topol. Géom. Différ. Catég. 29, 157-61 (1988)
  15. Joyal, A., Tierney, M.: An extension of the Galois theory of Grothendieck. Mem. Am. Math. Soc. 51(309), vii+71 (1984)
  16. Kock, A.: Limit Monads in Categories. Ph.D. Thesis, University of Chicago (1967)
  17. MacLane, S.: Categories for the Working Mathematician. Springer, New York (1971)
  18. Picado, J.: Weil Entourages in Pointfree Topology. Ph.D. Thesis, University of Coimbra (1995)
  19. Picado, J.: Weil uniformities for frames. Comment. Math. Univ. Carol. 36, 357-70 (1995)
  20. Picado, J.: Structured frames by Weil entourages. Appl. Categ. Struct. 8, 351-66 (2000) CrossRef
  21. Pultr, A.: Pointless uniformities I,II. Comment. Math. Univ. Carol. 25, 91-04, 105-20 (1984)
  22. Pultr, A.: Frames. In: Hazewinkel, M. (ed.) Handbook of Algebra, vol. 3, pp. 791-58. Elsevier, Amsterdam (2003) CrossRef
  23. Tukey, J.W.: Convergence and uniformity in topology. In: Annals of Mathematics Studies, vol. 2. Princeton University Press, Princeton (1940)
  24. Weil, A.: Sur les Espaces à Structure Uniforme et sur la Topologie Générale. Publications de l’Institute Mathématique de l’Université de Strasbourg, Hermann, Paris (1938)
  
• 作者单位：Jorge Picado (1)
  Ale? Pultr (2)
  
  1. CMUC, Department of Mathematics, University of Coimbra, 3001-454, Coimbra, Portugal
  2. Department of Applied Mathematics and ITI, MFF, Charles University, Praha, Czech Republic
  
• ISSN：1572-9095

文摘

Due to the nature of product in the category of locales, the entourage uniformities in the point-free context only mimic the classical Weil approach while the cover (Tukey type) ones can be viewed as an immediate extension. Nevertheless the resulting categories are concretely isomorphic. We present a transparent construction of this isomorphism, and apply it to the natural uniformities of localic groups. In particular we show that localic group homomorphisms are uniform, thus providing natural forgetful functors from the category of localic groups into any of the two categories of uniform locales.