More on Compact Hausdorff Spaces and Finitary Duality

Abstract

It is an old conjecture by P. Bankston that the category CompHaus of compact Hausdorff spaces and their continuous maps is not dually equivalent to any elementary P-class of finitary algebras (taken as a category with all homomorphisms between its members as maps), where elementary means defined by first order axioms, and a P-class is one closed under arbitrary (cartesian) products. One motivation for this conjecture is the fact that such a dual equivalence would make ultracopowers of compact Hausdorff spaces correspond to ultrapowers of finitary algebras, and one might expect this to have contradictory consequences.As a possible step towards proving his conjecture, Bankston [2] showed that no elementary SP-class of finitary algebras can be dually equivalent to CompHaus. However, it was subsequently proved in [1] that the same holds for any SP-class of finitary algebras, using an argument independent of ultrapowers.