Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Groups of dualities


Authors: Georgi D. Dimov and Walter Tholen
Journal: Trans. Amer. Math. Soc. 336 (1993), 901-913
MSC: Primary 18A40; Secondary 18D05, 22D35, 54B30, 54H10
DOI: https://doi.org/10.1090/S0002-9947-1993-1100693-0
MathSciNet review: 1100693
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For arbitrary categories $ \mathcal{A}$ and $ \mathcal{B}$ , the "set" of isomorphism-classes of dualities between $ \mathcal{A}$ and $ \mathcal{B}$ carries a natural group structure. In case $ \mathcal{A}$ and $ \mathcal{B}$ admit faithful representable functors to Set, this structure can often be described quite concretely in terms of "schizophrenic objects" (in the sense of Johnstone's book on "Stone Spaces"). The general theory provided here allows for a concrete computation of that group in case $ \mathcal{A} = \mathcal{B} = \mathcal{C}$ is the category of all compact and all discrete abelian groups: it is the uncountable group of algebraic automorphisms of the circle $ \mathbb{R}/\mathbb{Z}$ , modulo its subgroup $ {\mathbb{Z}_2}$ of continuous automorphisms.


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

  • [1] B. Banaschewski, More on compact Hausdorff spaces and finitary duality, Canad. J. Math. 36 (1984) 1113-1118. MR 771930 (86h:18002)
  • [2] G. M. Bergman, On the scarcity of contravariant left adjunctions, Algebra Universalis 24 (1987), 169-185. MR 921541 (88k:18003)
  • [3] E. Binz, Continuous convergence on $ C(X)$, Lecture Notes in Math., vol. 469, Springer-Verlag, 1975. MR 0461418 (57:1403)
  • [4] B. J. Day, Onto Gelfand transformations, Cahiers Topologie Géom. Différentielle Catégoriques 28 (1987), 227-234. MR 923630 (89e:18015)
  • [5] D. Dikranjan and A. Orsatti, On an unpublished manuscript of Ivan Prodanov concerning locally compact modules and their dualities, Comm. Algebra 17 (1989), 2739-2771. MR 1025606 (92a:13021)
  • [6] G. D. Dimov and W. Tholen, A characterization of representable dualities, Proc. Internat. Conf. on Categorical Topology (Prague, 1988), World Scientific, Singapore, 1989, pp. 336-357. MR 1047910 (91e:18003)
  • [7] C. Faith, Algebra. I. Rings, modules and categories, corrected reprint, Springer-Verlag, 1981. MR 623254 (82g:16001)
  • [8] P. J. Freyd, Algebra valued functors in general and tensor products in particular, Colloq. Math. 14 (1966), 89-106. MR 0195920 (33:4116)
  • [9] L. Fuchs, Infinite Abelian groups, vols. I and II, Academic Press, London, 1970. MR 0255673 (41:333)
  • [10] I. Glicksberg, Uniform boundedness for groups, Canad. J. Math. 14 (1962), 209-276. MR 0155923 (27:5856)
  • [11] H. Herrlich and G. E. Strecker, Category theory, 2nd ed., Heldermann Verlag, Berlin, 1979. MR 571016 (81e:18001)
  • [12] K. H. Hofmann and K. Keimel, A generalized character theory of partially ordered sets and lattices, Mem. Amer. Math. Soc. No. 122 (1972). MR 0340129 (49:4885)
  • [13] J. R. Isbell, General functorial semantics, I, Amer. J. Math. 94 (1972), 535-596. MR 0396718 (53:580)
  • [14] P. T. Johnstone, Stone spaces, Cambridge Univ. Press, Cambridge, 1982. MR 698074 (85f:54002)
  • [15] J. Lambek and B. A. Rattray, A general Stone-Gelfand duality, Trans. Amer. Math. Soc. 248 (1979), 1-35. MR 521691 (80i:18012)
  • [16] S. Mac Lane, Categories for the working mathematician, Springer-Verlag, 1971. MR 1712872 (2001j:18001)
  • [17] L. D. Nel, Topological universes and smooth Gelfand-Naimark duality, Contemp. Math., vol. 30, Amer. Math. Soc., Providence, R. I., 1984, pp. 244-276. MR 749775 (86b:18007)
  • [18] V. V. Pashenkov, Homogeneous and non-homogeneous duality, Uspekhi Mat. Nauk 42:5 (1987), 79-99; English transl. Russian Math. Surveys 42:5 (1987), 95-121.
  • [19] H.-E. Porst and M. B. Wischnewsky, Every topological category is convenient for Gelfand duality, Manuscripta Math. 25 (1978), 169-204. MR 0491873 (58:11060)
  • [20] I. Prodanov, An axiomatic characterization of the Pontryagin duality, unpublished manuscript (Sofia, 1984).
  • [21] A Pultr, The right adjoints into the category of relational systems, Lecture Notes in Math., vol. 137, Springer-Verlag, 1970. MR 0263897 (41:8496)
  • [22] D. W. Roeder, Functorial characterizations of Pontryagin duality, Trans. Amer. Math. Soc. 154 (1971), 151-175. MR 0279233 (43:4956)
  • [23] N. Varopoulos, Studies in harmonic analysis, Proc. Cambridge Philos. Soc. 60 (1964), 465-516. MR 0163985 (29:1284)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 18A40, 18D05, 22D35, 54B30, 54H10

Retrieve articles in all journals with MSC: 18A40, 18D05, 22D35, 54B30, 54H10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1100693-0
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society