What is the directed interval 2?

It is the 'walking arrow,' a primitive representing a single directed morphism from 0 to 1, crucial for modeling time-sensitive processes where reversal is not possible.

How does this differ from standard cubical type theory?

Standard cubical type theory uses undirected intervals (𝕀); this skill introduces directedness, meaning there is a morphism from 0 to 1, but specifically NOT from 1 to 0.

What languages does this skill support?

It is designed to work with formal languages such as Rzk and Cubical Agda, which are used for synthetic ∞-category theory and directed type theory.

What is a 'chemputer' in this context?

A chemputer is a topological model where chemical reactions are represented as paths within directed cubes, allowing for formal verification of complex chemical synthesis.

Can this be used for standard software state machines?

Yes, it is highly effective for modeling any system where state transitions are irreversible or where the 'arrow of time' must be strictly enforced in the logic.

Directed Interval Modeling

Name: Directed Interval Modeling
Author: plurigrid

byplurigrid

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Developer Tools

Axiomatizes time-directed homotopy and reaction progress using directed intervals for chemical and topological computations.

The Directed Interval skill implements the 'walking arrow' (0 → 1) framework based on Riehl-Shulman's synthetic ∞-category theory. It enables Claude to model irreversible processes, such as chemical reactions and time-flowing state transitions, using Rzk and Cubical Agda. By replacing undirected intervals with directed ones, this skill provides a robust foundation for modeling 'chemputer' semantics, higher simplices for multi-step syntheses, and extension types for formal verification of time-dependent workflows where 0 necessarily leads to 1 without a return path.

Key Features

01Modeling of irreversible 0 → 1 state transitions (the 'walking arrow')

02Support for extension types to define functions with prescribed boundaries

03Implementation of the Riehl-Shulman directed cube category

04Chemputer semantics for mapping topological cubes to chemical reaction stages

057 GitHub stars

06Integration with Rzk and Cubical Agda for formal process verification

Use Cases

01Formalizing chemical reaction pathways as directed hom types

02Modeling irreversible state transitions in distributed systems or 'chemputers'

03Verifying multi-step synthesis logic using higher simplices and Δⁿ shapes

What are Skills?·How to Install

Install with 🐟 Skill.Fish

npx skillfish add plurigrid/asi directed-interval

For use in Claude.ai and ChatGPT

Download Skill