Stone Duality Above Dimension Zero: Axiomatising the Algebraic Theory of C(X)
Stone duality above dimension zero: Axiomatising the algebraic theory of C(X) Vincenzo Marraa, Luca Reggioa,b aDipartimento di Matematica \Federigo Enriques", Universit`adegli Studi di Milano, via Cesare Saldini 50, 20133 Milano, Italy bUniversit´eParis Diderot, Sorbonne Paris Cit´e,IRIF, Case 7014, 75205 Paris Cedex 13, France Abstract It has been known since the work of Duskin and Pelletier four decades ago that Kop, the opposite of the category of compact Hausdorff spaces and continuous maps, is monadic over the category of sets. It follows that Kop is equivalent to a possibly infinitary variety of algebras ∆ in the sense of S lomi´nskiand Linton. Isbell showed in 1982 that the Lawvere-Linton algebraic theory of ∆ can be generated using a finite number of finitary operations, together with a single operation of countably infinite arity. In 1983, Banaschewski and Rosick´yindependently proved a conjecture of Bankston, establishing a strong negative result on the axiomatisability of Kop. In particular, ∆ is not a finitary variety | Isbell's result is best possible. The problem of axiomatising ∆ by equations has remained open. Using the theory of Chang's MV-algebras as a key tool, along with Isbell's fundamental insight on the semantic nature of the infinitary operation, we provide a finite axiomatisation of ∆. Key words: Algebraic theories, Stone duality, Boolean algebras, MV-algebras, Lattice-ordered Abelian groups, C∗-algebras, Rings of continuous functions, Compact Hausdorff spaces, Stone-Weierstrass Theorem, Axiomatisability. 2010 MSC: Primary: 03C05. Secondary: 06D35, 06F20, 46E25. 1. Introduction. In the category Set of sets and functions, coordinatise the two-element set as 2 := f0; 1g.
[Show full text]