CAT# - Category Theory Tripartite Skill

Three worldlines, one universal property

Core Triad

SICP (-1) ⊕ CTP (0) ⊕ CatColab (+1) = 0 mod 3 ✓

Worldline	Language	Dispatch	Order Problem	Resolution
-1 SICP	Scheme	(put 'op '(t1 t2) proc)	Coercion tower fixed	Explicit hierarchy
0 CTP	Racket	(category-compose C f g)	Functors associative	Natural transformations
+1 CatColab	Julia/Rust	elaborate_model(theory)	Concurrent edits	Automerge CRDT

Worldline -1: SICP Generic Operations

From SICP §2.5 + SDF §3.1:

;; Data-directed dispatch table
(put 'add '(complex complex)
  (lambda (z1 z2) (tag 'complex (add-complex z1 z2))))

;; Coercion tower (ORDER MATTERS)
(put-coercion 'scheme-number 'complex scheme-number->complex)

;; SDF extension pattern
(define (extend-generic-arithmetic! generic-arithmetic extender)
  (add-to-generic-arithmetic! generic-arithmetic
    (extender generic-arithmetic)))

;; The problem: "construction is by assignment"
(extend-generic-arithmetic! g symbolic-extender)
(extend-generic-arithmetic! g function-extender)  ; ORDER DEPENDENT

Key insight: set! introduces temporal ordering into what should be compositional.

Worldline 0: CTP Racket Categories

From NoahStoryM/ctp:

#lang racket
(require ctp)

;; Category as first-class value
(define-category Nat
  #:objects natural?
  #:morphisms (λ (a b) (and (natural? a) (natural? b) (<= a b)))
  #:identity (λ (a) a)
  #:compose (λ (f g) g))  ; transitivity

;; Functor: structure-preserving map
(define-functor F
  #:from Nat
  #:to Set
  #:object-map (λ (n) (range n))
  #:morphism-map (λ (f) (λ (s) (take s (length s)))))

;; Natural transformation: morphism of functors
(define-natural-transformation η
  #:from F
  #:to G
  #:component (λ (a) (λ (x) (list x))))

Key insight: Composition is associative by axiom—order independence built in.

Worldline +1: CatColab Double Theories

From CatColab catlog (Rust):

// Double theory: 2-categorical structure
pub trait DblTheory: VDblCategory {
    type ObType;   // Object generators
    type MorType;  // Morphism generators
    type ObOp;     // Object operations  
    type MorOp;    // Morphism operations
}

// Stock-Flow theory for epidemiology
pub fn th_stock_flow() -> DiscreteDblTheory {
    let mut cat = FpCategory::new();
    cat.add_ob_generator(name("Stock"));
    cat.add_mor_generator(name("Flow"), name("Stock"), name("Stock"));
    cat.add_mor_generator(name("Link"), name("Stock"), name("Stock"));
    cat.into()
}

// Extension via DPO rewriting (order-independent)
let extended = compose_theories(th_stock_flow(), th_vaccination());

Key insight: Automerge CRDT resolves concurrent extensions without ordering.

The Universal Property

All three worldlines converge on the same universal construction:

         extend
Base ──────────→ Extended
  │                 │
  │ U               │ U'
  ↓                 ↓
 Set ───────────→ Set
         F

The extension is universal iff the square commutes AND the extension is minimal.

Interleaving SDF + CTP

SDF §3.2 Predicate Dispatch → CTP Functor

;; SDF: predicate dispatch eliminates order dependence
(define (make-generic-operation name arity default-operation)
  (let ((rules '()))
    (define (add-rule! predicate handler priority)
      (set! rules (insert-by-priority rules predicate handler priority)))
    ...))

;; CTP equivalent: rules as morphisms in a category
(define-category DispatchRules
  #:objects type-predicate?
  #:morphisms (λ (p1 p2) (implies? p1 p2))  ; subsumption
  #:identity identity-predicate
  #:compose predicate-conjunction)

SDF §7 Propagators → CTP Limits

;; SDF: propagator as cell network
(define (make-propagator neighbors output-proc)
  ...)

;; CTP: propagator network as diagram, solution as limit
(define-diagram PropNet
  #:shape (category-of-graph propagator-graph)
  #:functor (λ (node) (cell-value node)))

(define solution (limit PropNet))  ; universal property!

AFL Parallel

Coverage-guided fuzzing shares structure with generic dispatch:

AFL                          Generic Arithmetic
────────────────────────────────────────────────
seed corpus            ↔     base arithmetic
mutation operators     ↔     extenders
coverage bitmap        ↔     dispatch store
path discovery         ↔     type combinations
mutation ORDER         ↔     extension ORDER
priority schedules     ↔     predicate priorities

UV One-Liner Bridge

# Python bridge to all three
uv run --no-project --with requests --with networkx python3 << 'EOF'
import networkx as nx

# Model the tripartite as a graph
G = nx.DiGraph()
G.add_edge("SICP", "CTP", weight=-1, label="lift dispatch to functor")
G.add_edge("CTP", "CatColab", weight=0, label="double theory")
G.add_edge("CatColab", "SICP", weight=+1, label="CRDT resolves order")

# GF(3) check
total = sum(d['weight'] for _, _, d in G.edges(data=True))
print(f"GF(3) sum: {total} mod 3 = {total % 3}")
assert total % 3 == 0, "Conservation violated!"
EOF

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End-of-Skill Interface

Commands

# Clone CTP
git clone https://github.com/noahstorym/ctp ~/worlds/ctp
cd ~/worlds/ctp && raco pkg install

# Run CTP examples
racket -l ctp -e '(require ctp/examples/nat)'

# Interleave with SDF
just cat-tripartite-sicp    # Extract SICP §2.5
just cat-tripartite-ctp     # Run CTP category examples
just cat-tripartite-catcolab # Start CatColab dev server

# Verify GF(3) conservation
just cat-tripartite-verify

References

SICP (-1)

• SICP Full Text
• SDF Extracted

CTP (0)

• CTP Racket Docs
• GitHub: NoahStoryM/ctp

CatColab (+1)

• CatColab Live
• catlog Rust Engine

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Skill: cat-tripartite
Trit: Balanced (contains all three)
GF(3): (-1) + (0) + (+1) = 0 ✓

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Autopoietic Marginalia

The interaction IS the skill improving itself.

Every use of this skill is an opportunity for worlding:

• MEMORY (-1): Record what was learned
• REMEMBERING (0): Connect patterns to other skills
• WORLDING (+1): Evolve the skill based on use

Add Interaction Exemplars here as the skill is used.