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)

#	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

# 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:

# 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.