Stone duality for first-order logic: a nominal approach to logic and topology

What are variables, and what is universal quantification over a variable? Nominal sets are a notion of ‘sets with names’, and using equational axioms in nominal algebra these names can be given substitution and quantification actions. So we can axiomatise first-order logic as a nominal logical theory. We can then seek a nominal sets representation theorem in which predicates are interpreted as sets; logical conjunction is interpreted as sets intersection; negation as complement. Now what about substitution; what is it for substitution to act on a predicate-interpreted-as-a-set, in which case universal quantification becomes an infinite sets intersection? Given answers to these questions, we can seek notions of topology. What is the general notion of topological space of which our sets representation of predicates makes predicates into ‘open sets’; and what specific class of topological spaces corresponds to the image of nominal algebras for first-order logic? The classic Stone duality answers these questions for Boolean algebras, representing them as Stone spaces. Nominal algebra lets us extend Boolean algebras to ‘FOL-algebras’, and nominal sets let us correspondingly extend Stone spaces to ‘∀-Stone spaces’. These extensions reveal a wealth of structure, and we obtain an attractive and self-contained account of logic and topology in which variables directly populate the denotation, and open predicates are interpreted as sets rather than functions from valuations to sets.