Cat Tripartite

---
name: cat-tripartite
description: 'Category Theory Tripartite: SICP generic dispatch, CTP Racket categories,
  CatColab double theories across 3 parallel worldlines with GF(3) conservation'
metadata:
  trit_minus: sicp
  trit_ergodic: ctp-racket
  trit_plus: catcolab
  seed: 1069
  interface_ports:
  - References
  - Commands
---
# 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:

```scheme
;; 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](https://github.com/noahstorym/ctp):

```racket
#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):

```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

```scheme
;; 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

```scheme
;; 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

```bash
# 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
```

---

## End-of-Skill Interface

## Commands

```bash
# 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](https://mitpress.mit.edu/sites/default/files/sicp/full-text/book/book.html)
- [SDF Extracted](file:///Users/bob/ies/sussman-hanson-software-design-flexibility.md)

### CTP (0)
- [CTP Racket Docs](https://docs.racket-lang.org/ctp/index.html)
- [GitHub: NoahStoryM/ctp](https://github.com/noahstorym/ctp)

### CatColab (+1)
- [CatColab Live](https://catcolab.org)
- [catlog Rust Engine](file:///Users/bob/worlds/T/CatColab/packages/catlog)

---

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


---

## 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.*