CTP-Yoneda Skill

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
Docs: docs.racket-lang.org/​ctp
Local: .topos/​ctp/

Chapters (GF(3) Colored)

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

# 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

Colored Awareness Protocol

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

# 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
• Qi Flow Language - Inspiration for CTP
• Category Theory in Context - Emily Riehl
• Category Theory for Computing Science - Barr & Wells
• nLab
• TheCatsters YouTube

Commands

# 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.