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 is one such and is the category of topological spaces, we consider a pair of adjoint functors which are represented by objects *I* and *J* in the sense that the underlying sets of and are and . (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 and that Fix *UF* = Image *U* = the limit closure of *J* in . For example, this happens if (1) is a limit closed subcategory of an equational category, (2) *J* is compact Hausdorff and has a basis of open sets of the form , where and are unary -operations, and (3) there are quaternary operations and such that, for all if and only if or . (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 , *n* finite, can be uniformly approximated arbitrarily closely by -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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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