What is a directed interval in this context?

A directed interval, often called the 'walking arrow,' is a formal structure representing a one-way path from 0 to 1. Unlike undirected intervals, it models processes where time or state flows forward and cannot be reversed.

Is this skill compatible with standard type theory?

Yes, it is designed to be compatible with cubical type theory machinery while specifically extending it to support directed (non-reversible) morphisms.

What are higher simplices like Δ² and Δ³ used for?

These higher-dimensional shapes model multi-step processes. For example, Δ² represents a two-step synthesis (A → B → C), and Δ³ represents a three-step synthesis with formal associativity.

How does this skill benefit chemical modeling?

It provides a mathematical foundation where 0 represents reactants and 1 represents products. This allows the 'chemputer' to treat chemical reactions as topological pathways that can be formally verified.

Directed Interval Type Theory

Name: Directed Interval Type Theory
Author: plurigrid

byplurigrid

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Data Science & ML

Axiomatizes the directed interval (0 → 1) to model irreversible processes, time-directed homotopy, and chemical reaction pathways.

The Directed Interval skill implements the 'walking arrow' concept from synthetic ∞-category theory, replacing undirected intervals with directed ones to represent irreversible flow. It provides a formal framework for modeling 'time-directed' computations, such as chemical reactions (reactants to products) or sequential workflows, using Rzk-based definitions. This skill is essential for developers working on topological compute systems, formal methods, or complex state transitions where directionality and extension types are critical for maintaining compositional coherence.

Key Features

01Axiomatizes the directed interval 2 (0 → 1) for time-irreversible modeling

02Implements extension types for partial elements and boundary prescriptions

03Provides integration patterns for Julia ACSets and SplitMix64 time axes

04Maps chemical semantics to directed cubes for reaction progress tracking

058 GitHub stars

06Defines higher directed cubes and simplices (Δ², Δ³) for multi-step processes

Use Cases

01Formalizing time-directed state transitions in topological computation systems

02Building synthetic ∞-category structures within Rzk or Julia environments

03Modeling irreversible chemical reaction pathways from reactants to products

What are Skills?·How to Install

Install with 🐟 Skill.Fish

npx skillfish add plurigrid/asi directed-interval

For use in Claude.ai and ChatGPT

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