Groups of dualities
HTML articles powered by AMS MathViewer
- by Georgi D. Dimov and Walter Tholen PDF
- Trans. Amer. Math. Soc. 336 (1993), 901-913 Request permission
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
- B. Banaschewski, More on compact Hausdorff spaces and finitary duality, Canad. J. Math. 36 (1984), no. 6, 1113–1118. MR 771930, DOI 10.4153/CJM-1984-063-6
- George M. Bergman, On the scarcity of contravariant left adjunctions, Algebra Universalis 24 (1987), no. 1-2, 169–185. MR 921541, DOI 10.1007/BF01188394
- Ernst Binz, Continuous convergence on $C(X)$, Lecture Notes in Mathematics, Vol. 469, Springer-Verlag, Berlin-New York, 1975. MR 0461418
- B. J. Day, Onto Gel′fand transformations, Cahiers Topologie Géom. Différentielle Catég. 28 (1987), no. 3, 227–234 (English, with French summary). MR 923630
- Dikran Dikranjan and Adalberto Orsatti, On an unpublished manuscript of Ivan Prodanov concerning locally compact modules and their dualities, Comm. Algebra 17 (1989), no. 11, 2739–2771. MR 1025606, DOI 10.1080/00927878908823873
- Georgi D. Dimov and Walter Tholen, A characterization of representable dualities, Categorical topology and its relation to analysis, algebra and combinatorics (Prague, 1988) World Sci. Publ., Teaneck, NJ, 1989, pp. 336–357. MR 1047910
- Carl Faith, Algebra. I. Rings, modules, and categories, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 190, Springer-Verlag, Berlin-New York, 1981. Corrected reprint. MR 623254
- P. Freyd, Algebra valued functors in general and tensor products in particular, Colloq. Math. 14 (1966), 89–106. MR 195920, DOI 10.4064/cm-14-1-89-106
- László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
- Irving Glicksberg, Uniform boundedness for groups, Canadian J. Math. 14 (1962), 269–276. MR 155923, DOI 10.4153/CJM-1962-017-3
- Horst Herrlich and George E. Strecker, Category theory, 2nd ed., Sigma Series in Pure Mathematics, vol. 1, Heldermann Verlag, Berlin, 1979. An introduction. MR 571016
- Karl Heinrich Hofmann and Klaus Keimel, A general character theory for partially ordered sets and lattices, Memoirs of the American Mathematical Society, No. 122, American Mathematical Society, Providence, R.I., 1972. MR 0340129
- John R. Isbell, General functorial semantics. I, Amer. J. Math. 94 (1972), 535–596. MR 396718, DOI 10.2307/2374638
- Peter T. Johnstone, Stone spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, Cambridge, 1982. MR 698074
- J. Lambek and B. A. Rattray, A general Stone-Gel′fand duality, Trans. Amer. Math. Soc. 248 (1979), no. 1, 1–35. MR 521691, DOI 10.1090/S0002-9947-1979-0521691-4
- Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
- L. D. Nel, Topological universes and smooth Gel′fand-Naĭmark duality, Mathematical applications of category theory (Denver, Col., 1983) Contemp. Math., vol. 30, Amer. Math. Soc., Providence, RI, 1984, pp. 244–276. MR 749775, DOI 10.1090/conm/030/749775 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.
- Hans-E. Porst and Manfred B. Wischnewsky, Every topological category is convenient for Gelfand duality, Manuscripta Math. 25 (1978), no. 2, 169–204. MR 491873, DOI 10.1007/BF01168608 I. Prodanov, An axiomatic characterization of the Pontryagin duality, unpublished manuscript (Sofia, 1984).
- A. Pultr, The right adjoints into the categories of relational systems, Reports of the Midwest Category Seminar, IV, Lecture Notes in Mathematics, Vol. 137, Springer, Berlin, 1970, pp. 100–113. MR 0263897
- David W. Roeder, Functorial characterizations of Pontryagin duality, Trans. Amer. Math. Soc. 154 (1971), 151–175. MR 279233, DOI 10.1090/S0002-9947-1971-0279233-X
- N. Th. Varopoulos, Studies in harmonic analysis, Proc. Cambridge Philos. Soc. 60 (1964), 465–516. MR 163985, DOI 10.1017/s030500410003797x
Additional Information
- © Copyright 1993 American Mathematical Society
- 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