Stone duality above dimension zero: Axiomatising the algebraic theory of C(X)

It has been known since the work of Duskin and Pelletier four decades ago that KH^op, the category opposite to compact Hausdorff spaces and continuous maps, is monadic over the category of sets. It follows that KH^op is equivalent to a possibly infinitary variety of algebras V in the sense of Slominski and Linton. Isbell showed in 1982 that the Lawvere-Linton algebraic theory of V can be generated using a finite number of finitary operations, together with a single operation of countably infinite arity. In 1983, Banaschewski and Rosicky independently proved a conjecture of Bankston, establishing a strong negative result on the axiomatisability of KH^op. In particular, V is not a finitary variety--Isbell's result is best possible. The problem of axiomatising V 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 V. Inter alia, we also give a short proof of a new negative result on the axiomatisability of KH^op, and we prove that any variety of algebras that is equivalent, as a category, to KH^op, must have no non-trivial proper sub varieties. The proof of our main result and its corollaries depend on the representation theory of MV-algebras--including the Cignoli-Dubuc-Mundici adjunction from 2004 between MV-algebras and KH^op, which we recall here for the reader's convenience--and on the Stone-Weierstrass Theorem.