Implement ordered locales with frames and preorders in Julia/Catlab workflows.
npx skills add https://github.com/plurigrid/asi --skill ordered-locale
Please help me install this Agent Skill. Skill: ordered-locale Source: https://github.com/plurigrid/asi/tree/main/skills/ordered-locale Command: npx skills add https://github.com/plurigrid/asi --skill ordered-locale
This Skill introduces ordered locales—frames enriched with a compatible preorder and the open cone condition—extending Stone duality to directed settings.
Run the demo commands (e.g., ordered-locale-demo) or verify the open cone condition with ordered-locale-verify; explore example locales like the Sierpinski or diamond locale.
High-intent search queries and answers about installing and using this skill.
Ordered locales combine frames with compatible preorders under the open cone condition. This Skill uses Catlab's ACSet schema to model ordered locales, treating frame operations as subobject Heyting algebras and ensuring upper and lower cones are open, enabling categorical topology workflows.
The open cone condition ensures that upper and lower cones in a preorder remain open under frame operations. This extends Stone duality to directed settings, making it essential for applications like spacetime causality, domain theory, and directed homotopy.
Yes. Ordered locales apply directly to spacetime causality analysis, domain theory, and modal logic by encoding preorder structure with open cone constraints. Catlab integration lets you verify these properties and compute limits and colimits for cone and cocone constructs.
Run the ordered-locale-verify command to check whether your frame and preorder satisfy the open cone condition. The Skill provides demo commands and example locales like Sierpinski and diamond to explore and validate implementations.
You need Julia with Catlab installed. Ordered locales leverage Catlab's ACSet framework to define frames, preorders, and cone constructs, so familiarity with categorical concepts and Catlab's schema system is beneficial.
Stone duality traditionally links Boolean algebras and compact Hausdorff spaces. Ordered locales extend this to directed topology by enriching frames with preorders and enforcing open cones, bridging classical duality and directed categorical structures.
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