CTP-Yoneda Skill Skill | Agent Skills

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):

#E67F86 (+1) - info.rkt
#D06546 (-1) - main.rkt
#1316BB (0) - ctp.scrbl
#BA2645 (+1) - category/main.scrbl
#49EE54 (-1) - functor/main.scrbl
#11C710 (0) - natural transformation/
#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.

Agent Skills: CTP-Yoneda Skill

Install this agent skill to your local

Skill Files