---
title: "Localic Esakia Duality via Conic Frames"
authors: ["Nesta van der Schaaf"]
year: 2026
venue: "arXiv:math.CT"
arxiv: "2608.26084"
doi: null
source: "https://arxiv.org/abs/2608.26084"
topics: ["curiosity", "maths-foundations"]
seed_rank: 1545
seed_batch: "curiosity-2026-09-03"
reviewed: "2026-09-03"
pool: "maths-foundations"
relevance_score: 10
lineage: pointfree-topology
cites:
  - title: "Localic Esakia Duality via Conic Frames"
    url: "https://arxiv.org/abs/2608.26084"
    year: 2026
    arxiv: "2608.26084"
    doi: null
see:
  - "1472-quantales-as-formal-concept-lattices"
---

# Localic Esakia Duality via Conic Frames

## One-sentence takeaway

Esakia duality goes fully constructive and localic: Heyting frames + conic frames recover the algebra without the Prime Ideal Theorem.

## Why it's lovely

Why you might love this: classical Esakia duals of Heyting algebras lean on the Prime Ideal Theorem — choice you cannot run in a constructive or type-theoretic setting. van der Schaaf rebuilds the dual on locales (Townsend-style Priestley) so the spectrum is a frame, not a space that needs ultrafilters. Logic that stays honest about what it assumes.

## Key ideas

- Classical Esakia: Heyting algebras ↔ Esakia spaces; recovery of the algebra uses PIT (non-constructive).
- Localic rewrite: Heyting frames on the algebraic side; conic frames as the dual geometric objects.
- Builds on Townsend's localic Priestley duality; the Esakia specialization is the new piece.
- Fully constructive: no ultrafilters, no Boolean Prime Ideal Theorem.

## Caveats

Localic duals are frames / locales, not spectral spaces with points; readers wanting classical point-set Esakia still need PIT. "Recently introduced Heyting frames" — check the cited source for the exact axiomatics.

## Implications for Broadside

Pairs with quantales-as-FCA (1472) on the pointfree shelf. Relevant whenever ano / Broadside wants Heyting or intuitionistic structure without smuggling classical choice into the semantics.

## Links

- arXiv abs: https://arxiv.org/abs/2608.26084
- PDF: https://arxiv.org/pdf/2608.26084.pdf