Choice-free topological duality for implicative lattices and Heyting algebras

We develop a common semantic framework for the interpretation both of \({\textbf {IPC}}\), the intuitionistic propositional calculus, and of logics weaker than \({\textbf {IPC}}\) (substructural and subintuitionistic logics). To this end, we prove a choice-free representation and duality theorem for implicative lattices, which may or may not be distributive. The duality specializes to a choice-free duality for the full subcategory of Heyting algebras and a category of topological sorted frames with a ternary sorted relation.