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

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

• Basic category theory primitives
• Functor composition
• Full Kan extension computation
• Adjunction verification
• Neural network parameter transfer demo