home / skills / plurigrid / asi / zigzag-equivalence

zigzag-equivalence skill

safe

This skill analyzes and refactors localization paths using zigzag equivalences to streamline routing and improve maintainability.

npx playbooks add skill plurigrid/asi --skill zigzag-equivalence

Review the files below or copy the command above to add this skill to your agents.

Files (2)

SKILL.md

828 B

---
name: zigzag-equivalence
description: "Zigzag equivalences: localization paths."
metadata:
  letter: Z
  trit: 0
  seed: 1159
---
# Z: Zigzag Equivalence

> *Zigzag equivalences: localization paths.*

## Directory Tree

```
zigzag-equivalence/
├── SKILL.md
├── hammock/
├── span-cospan.lean
```

## GF(3) Assignment

Letter Z → trit = 0

## Integration

```scheme
(define (zigzag_equivalence x)
  ;; Zigzag equivalences: localization paths.
  x)
```


---

## Autopoietic Marginalia

> **The interaction IS the skill improving itself.**

Every use of this skill is an opportunity for worlding:
- **MEMORY** (-1): Record what was learned
- **REMEMBERING** (0): Connect patterns to other skills  
- **WORLDING** (+1): Evolve the skill based on use



*Add Interaction Exemplars here as the skill is used.*

Overview

This skill captures zigzag equivalences as localization paths for topological and categorical workflows. It frames sequences of morphisms and localizations as concrete paths that can be inspected, compared, and composed. The goal is to make reasoning about equivalences and rewrites in zigzag diagrams practical for applications in topology, category theory, and computational chemistry workflows.

How this skill works

The skill inspects zigzag diagrams consisting of alternating spans and cospans, treating each segment as a localization step. It identifies when two diagrams define equivalent localized objects by tracking invertible moves and composition along the path. Outputs are compact summaries of equivalence classes and suggested simplifications or canonical representatives.

When to use it

When you need to decide if two localization sequences yield the same object or state.
When simplifying composite rewrites in categorical or topological models.
During the design or analysis of chemical computation pipelines modeled as topological processes.
When formalizing equivalences in homotopical or derived contexts where zigzag diagrams arise.
While validating transformations that alternate spans and cospans in a workflow.

Best practices

Represent each morphism and localization explicitly so the skill can trace invertible steps.
Keep diagrams small and modular; compose canonical representatives incrementally.
Annotate edges with roles (e.g., localization, inclusion, projection) to improve detection.
Use the skill iteratively: inspect, simplify, and re-inspect to converge on canonical forms.
Cross-check suggested equivalences against domain constraints (e.g., physical or chemical invariants).

Example use cases

Verify two different localization routes in a topological chemputer pipeline produce equivalent outputs.
Simplify a complex zigzag of spans and cospans into a minimal canonical path for documentation.
Compare alternative proofs or constructions in category-theoretic work that use zigzag homotopies.
Automate detection of redundant localizations in an experimental protocol modeled categorically.
Generate human-readable summaries of equivalence classes for review by domain experts.

FAQ

What kinds of diagrams does the skill accept?

It accepts alternating sequences of spans and cospans expressed as localization steps; explicit edge annotations improve accuracy.

Can it produce a canonical representative for an equivalence class?

Yes. It suggests simplified or canonical paths by removing invertible moves and consolidating adjacent localizations when legal.