First-order logical duality

From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, in this case 2. In the present work, we generalize the entire arrangement from propositional to first-order logic. Boolean algebras are replaced by Boolean categories presented by theories in first-order logic, and spaces of models are replaced by topological groupoids of models and their isomorphisms. A duality between the resulting categories of syntax and semantics, expressed first in the form of a contravariant adjunction, is established by homming into a common dualizing object, now Sets, regarded once as a boolean category, and once as a groupoid equipped with an intrinsic topology. ∗Corresponding author. Present address: Dept. of Informatics, University of Oslo, PO Box 1080 Blindern, N-0316 Oslo, Norway. Phone: (+47) 22 85 27 81. Fax: (+47) 22 85 24 01