Categorical Composition

# Categorical Composition

**Category:** Phase 3 Core - Compositional Architecture
**Status:** Skeleton Implementation
**Dependencies:** None (foundational)

## Overview

Implements categorical abstractions for compositional learning: Kan extensions for adapting between learning problems, higher adjunctions for bidirectional transformations, and functorial parameter transfer for compositional generalization.

## Capabilities

- **Kan Extensions**: Left/right Kan extensions for problem adaptation
- **Adjunctions**: Adjoint functors for bidirectional transformations
- **Functorial Transfer**: Preserve structure across parameter spaces
- **Compositional Architecture**: Build complex systems from simple components

## Core Components

1. **Category Theory Primitives** (`category_theory.jl`)
   - Category, functor, natural transformation definitions
   - Composition and identity laws
   - Diagram chasing utilities

2. **Kan Extensions** (`kan_extensions.jl`)
   - Left Kan extension (initial/colimit-based)
   - Right Kan extension (terminal/limit-based)
   - Pointwise computation formulas

3. **Adjunctions** (`adjunctions.jl`)
   - Adjoint functor pairs
   - Unit and counit natural transformations
   - Triangle identities verification

4. **Functorial Parameter Transfer** (`functorial_transfer.jl`)
   - Transfer neural network parameters via functors
   - Preserve compositional structure
   - Zero-shot generalization via categorical reasoning

## Integration Points

- **Input from**: All Phase 3 skills (provides compositional framework)
- **Output to**: All Phase 3 skills (foundational abstraction)
- **Coordinates with**: `formal-verification-ai` (correctness proofs)

## Usage

```julia
using CategoricalComposition

# Define source and target categories
C = FiniteCategory(objects=[:A, :B], morphisms=Dict(:f => (:A, :B)))
D = FiniteCategory(objects=[:X, :Y, :Z], morphisms=Dict(:g => (:X, :Y), :h => (:Y, :Z)))

# Define functor F: C -> D
F = Functor(
    source=C,
    target=D,
    object_map=Dict(:A => :X, :B => :Y),
    morphism_map=Dict(:f => :g)
)

# Compute left Kan extension
G = Functor(source=C, target=Set, object_map=Dict(:A => [1,2], :B => [3,4]))
Lan_F_G = left_kan_extension(F, G)

# Verify adjunction
@assert check_adjunction(Lan_F_G, restriction_functor(F))
```

## References

- Mac Lane "Categories for the Working Mathematician" (1971)
- Fong & Spivak "Seven Sketches in Compositionality" (2019)
- Shiebler et al. "Category Theory in Machine Learning" (2021)

## Implementation Status

- [x] Basic category theory primitives
- [x] Functor composition
- [ ] Full Kan extension computation
- [ ] Adjunction verification
- [ ] Neural network parameter transfer demo