Geometric and Higher Order Logic in terms of Abstract Stone Duality
Paul J. Taylor · Theory and applications of categories · 2000
The contravariant powerset, and its generalisations X to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that ^ F () = ^ F (>). Conversely, when the adjunction ( ) a ( ) is monadic, this equation implies that classifies some class of monos, and the Frobenius law 9x:((x) ^ ) = (9x:(x)) ^ ) for the existential quantier. In topology, the lattice duals of these equations also hold, and are related to the Phoa principle in synthetic domain theory. The natural definitions of discrete and Hausdorff spaces correspond to equality and inequality, whilst the quantifiers considered as adjoints characterise open (or, as we call them, overt) and compact spaces. Our treatment of overt discrete spaces and open maps is precisely dual to that of compact Hausdorff spaces and proper maps. The category of overt discrete ...