A point-free rewrite
An arXiv preprint presents a constructive, point-free route through Esakia duality, a formal correspondence built around Esakia frames and locales. The paper asks whether that duality can be recast without classical assumptions and whether Townsend’s localic Priestley duality can be restricted and factorized through Esakia frames. The authors state that the development avoids both the law of excluded middle and non-constructive choice principles.
The study analyzes formal point-free structures—Heyting frames, Esakia locales and Esakia frames—rather than an empirical population. Its conclusions therefore concern relationships within those mathematical structures, not measurements from people or a sampled dataset.
The cone behind the construction
At the center is a conic frame: a frame carrying a join-preserving cone. The cone is paired with a localic relation whose source is open, and the paper uses a cone–rel adjunction to move between the operation and the relation. The construction gives the paper’s order-theoretic and algebraic structures two linked descriptions while remaining point-free.
The framework includes two key recovery results. For every conic frame, the induced relation has an open source and recovers the original cone. In the other direction, closed localic relations with open source are fixed points of the cone–rel adjunction: translating the relation into cone data and back leaves the relation unchanged.
Turning frames into algebra
The paper defines an Esakia frame as a conic Stone frame with three extra features: its cone is a closure operator, it satisfies a closedness condition, and its compact elements are Boolean-generated by compact upper elements. These conditions specify how the frame’s order and algebraic structure fit together.
From there, the authors build the algebraic side. They show that the compact upper elements of an Esakia frame form a Heyting algebra. Its implication is defined through the cone: take the meet of the first element with the negation of the second, apply the cone, and then take negation again.
The construction also runs in reverse. Starting with a Heyting algebra, the paper produces an Esakia frame, written (EA, ▽A), and then shows that the compact-upper-element and patch constructions establish an equivalence between Esakia frames and Heyting algebras. The two settings can therefore be recovered from one another through the stated constructions.
A route through Townsend’s duality
The same framework connects to Heyting frames, another formal setting in the paper. The authors show that the stated functors establish an equivalence between Esakia frames and Heyting frames. They also show that Townsend’s localic Priestley duality restricts to a duality between Heyting frames and Esakia locales.
That restriction is not merely a parallel result. The restricted Townsend route agrees with the route through Esakia frames: both Townsend functors factor through the conic-frame constructions. The agreement places the two routes within the same formal framework.
What the result establishes
Taken together, the results establish a dual equivalence between Esakia frames and Esakia locales, an equivalence between Esakia frames and Heyting algebras, and an equivalence between Esakia frames and Heyting frames. The authors present this as a fully constructive localic Esakia duality, with conic frames clarifying the interaction between order and algebra.
The supplied document is an arXiv preprint, version one, dated 26 Aug 2026, and its subject is a family of point-free mathematical structures rather than an empirical population. The work reports partial funding from the French National Research Agency under the Plan France 2030 framework.
Paper data and sources
Original title: Localic Esakia Duality via Conic Frames
Authors: Nesta van der Schaaf
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text