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A general Stone-Gel'fand duality


Authors: J. Lambek and B. A. Rattray
Journal: Trans. Amer. Math. Soc. 248 (1979), 1-35
MSC: Primary 18C10; Secondary 46M15, 54B30
DOI: https://doi.org/10.1090/S0002-9947-1979-0521691-4
MathSciNet review: 521691
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Abstract: We give a simple characterization of full subcategories of equational categories. If $ \mathcal{a}$ is one such and $ \mathcal{B}$ is the category of topological spaces, we consider a pair of adjoint functors $ {\mathcal{a}^{op}}\underset{F}{\overset{U}{\longleftrightarrow}}\mathcal{B}$ which are represented by objects I and J in the sense that the underlying sets of $ U(A)$ and $ F(B)$ are $ \mathcal{a}(A,I)$ and $ \mathcal{B}(B,J)$. (One may take I and J to have the same underlying set.) Such functors always establish a duality between Fix FU and Fix UF. We study conditions under which one can conclude that FU and UF are reflectors into Fix FU and Fix UF, that Fix FU = Image F = the limit closure of I in $ \mathcal{a}$ and that Fix UF = Image U = the limit closure of J in $ \mathcal{B}$. For example, this happens if (1) $ \mathcal{a}$ is a limit closed subcategory of an equational category, (2) J is compact Hausdorff and has a basis of open sets of the form $ \{ x \in J\vert\alpha (I)(x) \ne \beta (I)(x)\} $, where $ \alpha$ and $ \beta $ are unary $ \mathcal{a}$-operations, and (3) there are quaternary operations $ \xi $ and $ \eta $ such that, for all $ x \in {J^4},\xi (I)(x) = \eta (I)(x)$ if and only if $ {x_1} = {x_2}$ or $ {x_3} = {x_4}$. (The compactness of J may be dropped, but then one loses the conclusion that Fix FU is the limit closure of I.) We also obtain a quite different set of conditions, a crucial one being that J is compact and that every f in $ \mathcal{B}({J^n},J)$, n finite, can be uniformly approximated arbitrarily closely by $ \mathcal{a}$-operations on I. This generalizes the notion of functional completeness in universal algebra. The well-known dualities of Stone and Gelfand are special cases of both situations and the generalization of Stone duality by Hu is also subsumed.


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

  • [1] R. F. Arens and I. Kaplansky, Topological representations of algebras, Trans. Amer. Math. Soc. 63 (1948), 457-481. MR 0025453 (10:7c)
  • [2] B. Banaschewski and E. Nelson, Boolean powers as algebras of continuous functions, McMaster Univ., 1976 (manuscript). MR 0429693 (55:2704)
  • [3] G. Birkhoff, Lattice theory, Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R. I., 1967. MR 0227053 (37:2638)
  • [4] P. Freyd, Algebra valued functors in general and tensor products in particular, Colloq. Math. 14 (1966), 89-106. MR 0195920 (33:4116)
  • [5] L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, Princeton, N. J., 1960. MR 0116199 (22:6994)
  • [6] G. Grätzer, Universal algebra, Van Nostrand, Princeton, N. J., 1968. MR 0248066 (40:1320)
  • [7] K. H. Hofmann and K. Keimel, A generalized character theory for partially ordered sets and lattices, Mem. Amer. Math. Soc. No 122 (1972). MR 0340129 (49:4885)
  • [8] T.-K. Hu, Stone duality for primal algebra theory, Math. Z. 110 (1969), 180-198. MR 0244130 (39:5447)
  • [9] -, On the topological duality for primal algebra theory, Algebra Universalis 1 (1971), 152-154. MR 0294218 (45:3291)
  • [10] J. R. Isbell, Top and its adjoint relatives, General Topology and Its Relation to Modern Algebra (Proc. Kanpur Topological Conf.), 1968, pp. 143-154.
  • [11] -, General functorial semantics. I, Amer. J. Math. 94 (1972), 535-596. MR 0396718 (53:580)
  • [12] K. Keimel and H. Werner, Stone duality for varieties generated by quasiprimal algebras, Mem. Amer. Math. Soc. No. 148 (1975), pp. 59-85. MR 0360411 (50:12861)
  • [13] L. Kučera and Z. Hedrlin, A universal extension theorem for structures and full embedding theorem for categories, Prague, 1969 (manuscript).
  • [14] J. Lambek and B. A. Rattray, Localization at injectives in complete categories, Proc. Amer. Math. Soc. 41 (1973), 1-9. MR 0414651 (54:2750)
  • [15] -, Localization and sheaf reflectors, Trans. Amer. Math. Soc. 210 (1975), 279-293. MR 0447364 (56:5677)
  • [16] -, Localization and duality in additive categories, Houston J. Math. 1 (1975), 87-100. MR 0404389 (53:8191)
  • [17] -, Functional completeness and Stone duality, Advances in Math. Suppl. Studies 1 (1978), 1-9. MR 520551 (80c:18006)
  • [18] F. W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 869-872. MR 0158921 (28:2143)
  • [19] F. E. J. Linton, Some aspects of equational categories (Proc. Conf. Categorical Algebra, La Jolla, 1965), Springer-Verlag, Berlin and New York, 1966, 84-94. MR 0209335 (35:233)
  • [20] -, Applied functorial semantics. I, Ann. Mat. Pura Appl. (4) 86 (1970), 1-13. MR 0285583 (44:2801)
  • [21] L. H. Loomis, An introduction to abstract harmonic analysis, Van Nostrand, Princeton, N. J., 1953. MR 0054173 (14:883c)
  • [22] S. Mac Lane, Categories for the working mathematician, Springer-Verlag, Berlin and New York, 1971. MR 0354798 (50:7275)
  • [23] J. M. Negrepontis, Duality in analysis from the point of view of triples, J. Algebra 19 (1971), 228-253. MR 0280571 (43:6291)
  • [24] J. Wick Pelletier, Examples of localizations, Comm. Algebra 3 (1975), 81-93. MR 0376803 (51:12978)
  • [25] A. Pultr, The right adjoints into the categories of relational systems, Lecture Notes in Math., vol. 137, Springer-Verlag, Berlin and New York, 1970. MR 0263897 (41:8496)
  • [26] G. E. Rickart, General theory of Banach algebras, Van Nostrand, Princeton, N. J., 1960. MR 0115101 (22:5903)
  • [27] H. Werner, Eine Charakterisierung functional vollständiger Algebren, Arch. Math. 21 (1970), 381-385. MR 0269574 (42:4469)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0521691-4
Keywords: Stone and Gelfand duality, operational and equational categories, adjoint functors, topological and uniform algebras, Weierstrass theorem and functional completeness
Article copyright: © Copyright 1979 American Mathematical Society

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