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)

#E67F86

#D06546

#1316BB

#BA2645

#49EE54

#11C710

#76B0F0

#E59798

#5333D9

#7E90EB

#1D9E7E

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