# Techniques: Ordered Locales

## What the paper contributes (one paragraph)

This is a pure mathematics paper in point-free topology and category theory. It extends Stone duality, the correspondence between topological spaces and locales, to take a preorder into account. The authors lift a preorder on the points of a topological space to a lower/upper/Egli-Milner preorder on its open subsets, axiomatise the result as a new category of "ordered locales", and restrict to spaces satisfying an "open cone" condition. They prove an adjunction between preordered topological spaces with open cones and enough points on one side, and ordered locales with monotone morphisms on the other, and show this adjunction restricts to an equivalence between sober T_0-ordered spaces with open cones and spatial ordered locales. The stated motivation is generalising ordered spaces into the point-free realm, with speculative future links to causality in quantum gravity. Nothing in the paper concerns quantum computation, hardware, or error correction.

## Techniques offered
- **order-extended Stone duality (ordered-locale adjunction)** - A category-theoretic method that axiomatises ordered locales and establishes an adjunction, restricting to an equivalence, between preordered topological spaces with open cones and spatial ordered locales.
  - guarantee: static categorical proof (an adjunction, restricting to an equivalence of categories); no statistical, operational, or computational assurance
  - quote: "We extend the Stone duality between topological spaces and locales to include order: there is an adjunction between the category of preordered topological spaces satisfying the so-called \emph{open cone} condition, and the newly defined category of \emph{ordered locales}."

## Where it could apply
- Target primitive(s): none
- Target application group(s): none
- Code family / hardware assumptions: none

## Caveats

Nothing in this paper is suppliable to an external vendor or application result in the pathfinder sense. It is abstract point-free topology and category theory (Stone-type duality extended with order), with no accreditation, decoding, fault-tolerant code, compilation, lattice surgery, error mitigation, benchmarking, characterisation, resource estimation, classical simulation, blind or delegated computation, privacy, secure computation, networking, or quantum algorithm. The only physics connection is speculative future work on a "pointfree approach to causality in quantum gravity", which is not quantum computation. The single bullet above is the best-fit method the paper genuinely supplies (a new mathematical duality result), recorded for completeness rather than because it maps onto a quantum-computing supply primitive.