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Transactions of the American Mathematical Society

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



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
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.

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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