---
name: ctp-yoneda
description: CTP-Yoneda Skill
version: 1.0.0
---

# CTP-Yoneda Skill

> *"The Yoneda lemma is arguably the most important result in category theory."*
> — Emily Riehl

Category Theory in Programming (CTP) by NoahStoryM - Racket tutorial mapping abstract CT concepts to programming constructs with GF(3) colored awareness.

## Overview

**Source**: [NoahStoryM/ctp](https://github.com/NoahStoryM/ctp)  
**Docs**: [docs.racket-lang.org/ctp](https://docs.racket-lang.org/ctp/index.html)  
**Local**: `.topos/ctp/`

## Chapters (GF(3) Colored)

| # | Chapter | Trit | Color | Status |
|---|---------|------|-------|--------|
| 1 | Category | +1 | `#E67F86` | ✓ Complete |
| 2 | Functor | -1 | `#D06546` | ✓ Complete |
| 3 | Natural Transformation | 0 | `#1316BB` | ✓ Complete |
| 4 | Yoneda Lemma | +1 | `#BA2645` | Planned |
| 5 | Higher Categories | -1 | `#49EE54` | Planned |
| 6 | (Co)Limits | 0 | `#11C710` | Planned |
| 7 | Adjunctions | +1 | `#76B0F0` | Planned |
| 8 | (Co)Monads | -1 | `#E59798` | Planned |
| 9 | CCC & λ-calculus | 0 | `#5333D9` | Planned |
| 10 | Toposes | +1 | `#7E90EB` | Planned |
| 11 | Kan Extensions | -1 | `#1D9E7E` | Planned |

**GF(3) Sum**: (+1) + (-1) + (0) + (+1) + (-1) + (0) + (+1) + (-1) + (0) + (+1) + (-1) = 0 ✓ BALANCED

## Core Concepts

### Category (Chapter 1)
- Objects, morphisms, composition, identity
- Digraphs → Free categories
- Subcategories, product/coproduct categories
- Quotient categories, congruence relations

### Functor (Chapter 2)  
- Structure-preserving maps between categories
- Constant, opposite, binary functors
- Hom functors (covariant/contravariant)
- Free monoid/category functors
- Finite automata as functors (DFA, NFA, TDFA)

### Natural Transformation (Chapter 3)
- Morphisms between functors
- Functor categories
- Vertical/horizontal composition
- Whiskering

### Yoneda Lemma (Key Insight)
```
Nat(Hom(A, -), F) ≅ F(A)
```
Every object is completely determined by its relationships to all other objects.

## Code Examples

Located in `.topos/ctp/scribblings/code/`:

### Category Examples
- `Set.rkt` - Category of sets
- `Rel.rkt` - Category of relations  
- `Proc.rkt` - Category of procedures
- `Pair.rkt` - Product category
- `Matr.rkt` - Matrix categories
- `List.rkt` - List monoid as category
- `Nat.rkt` - Natural numbers

### Functor Examples
- `DFA.rkt` - Deterministic finite automata
- `NFA.rkt` - Nondeterministic finite automata
- `TDFA.rkt` - Typed DFA
- `Set->Rel.rkt` - Set to Relation functor
- `P_*.rkt`, `P^*.rkt`, `P_!.rkt` - Powerset functors
- `SliF.rkt`, `coSliF.rkt` - Slice functors

## Racket Integration

```bash
# Install CTP package
cd .topos/ctp && raco pkg install

# Build documentation
raco setup --doc-index ctp

# Open docs
open doc/ctp/index.html
```

## Connection to Music-Topos

| CTP Concept | Music-Topos Implementation |
|-------------|---------------------------|
| Category | ACSets schema |
| Functor | Geometric morphism |
| Natural Transformation | Schema migration |
| Yoneda | Representable presheaves |
| Limits | Pullbacks in DuckDB |
| Adjunctions | Galois connections |
| Monads | Computation contexts |

## Colored Awareness Protocol

When reading CTP files, each touched file gets a deterministic color:

```ruby
# Track file access with Gay.jl colors
seed = 1069
files_touched = []

def touch_file(path, index)
  color = gay_color_at(seed, index)
  files_touched << { path: path, color: color, trit: color[:trit] }
end
```

Current session colors (seed=1069):
1. `#E67F86` (+1) - info.rkt
2. `#D06546` (-1) - main.rkt  
3. `#1316BB` (0) - ctp.scrbl
4. `#BA2645` (+1) - category/main.scrbl
5. `#49EE54` (-1) - functor/main.scrbl
6. `#11C710` (0) - natural transformation/
7. `#76B0F0` (+1) - code examples

## References

- [CTP Tutorial](https://docs.racket-lang.org/ctp/index.html)
- [Qi Flow Language](https://github.com/drym-org/qi) - Inspiration for CTP
- *Category Theory in Context* - Emily Riehl
- *Category Theory for Computing Science* - Barr & Wells
- [nLab](https://ncatlab.org/nlab/show/HomePage)
- [TheCatsters YouTube](https://www.youtube.com/@TheCatsters)

## Commands

```bash
# View CTP docs
just ctp-docs

# Run CTP examples
just ctp-examples

# Verify GF(3) coloring
just ctp-colors
```



## Scientific Skill Interleaving

This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:

### Graph Theory
- **networkx** [○] via bicomodule
  - Universal graph hub

### Bibliography References

- `category-theory`: 139 citations in bib.duckdb

## Cat# Integration

This skill maps to **Cat# = Comod(P)** as a bicomodule in the equipment structure:

```
Trit: 0 (ERGODIC)
Home: Presheaves
Poly Op: ⊗
Kan Role: Adj
Color: #26D826
```

### GF(3) Naturality

The skill participates in triads satisfying:
```
(-1) + (0) + (+1) ≡ 0 (mod 3)
```

This ensures compositional coherence in the Cat# equipment structure.