ctp-yoneda

# 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 ``` ## CT lattice atlas Part of: `para-mensch-commons` (CT lattice family).