Can I verify if a structure satisfies the open cone condition?

Yes, the skill includes built-in verification logic to ensure that the preorder and the topology interact correctly according to the Heunen-van der Schaaf definition.

What is an ordered locale?

An ordered locale is a locale (a point-free topological space or frame) equipped with a compatible preorder that satisfies the open cone condition, meaning its upper and lower cones are always open.

Is this skill suitable for physics modeling?

Yes, it is specifically designed for applications like spacetime causality where order represents causal influence and the point-free approach avoids traditional singularities.

How does this skill use Catlab.jl?

It leverages Catlab for schema definitions using ACSets and utilizes categorical theories to perform frame operations like meets and joins through universal constructions like limits and colimits.

Ordered Locale Point-Free Topology

Name: Ordered Locale Point-Free Topology
Author: plurigrid

byplurigrid

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Data Science & ML

Implements directed point-free topology using frames and preorders satisfying the open cone condition.

About

The Ordered Locale skill provides advanced mathematical tools for point-free topology, extending classical Stone duality to include directed order. It allows developers and researchers to model complex structures such as spacetime causality, domain theory, and directed homotopy using Julia’s Catlab.jl ecosystem. By implementing frames as complete Heyting algebras and enforcing the open cone condition, this skill enables rigorous verification of topological properties and categorical operations like limits and colimits within a computational environment.

Key Features

Frame-based meets, joins, and Heyting implications
2 GitHub stars
Categorical limits and colimits via Catlab.jl
Open cone condition verification (↑x and ↓x)
ACSet-based ordered locale schema implementation
MLX-accelerated finite locale approximations

Use Cases

Formal verification of directed paths in concurrent systems
Scott topology implementation for domain theory and computation
Causal coverage modeling for spacetime and physics simulations

About

Key Features

Frame-based meets, joins, and Heyting implications
2 GitHub stars
Categorical limits and colimits via Catlab.jl
Open cone condition verification (↑x and ↓x)
ACSet-based ordered locale schema implementation
MLX-accelerated finite locale approximations

Use Cases

Formal verification of directed paths in concurrent systems
Scott topology implementation for domain theory and computation
Causal coverage modeling for spacetime and physics simulations