A sleek and expressive Python library for modeling categories, functors, natural transformations, and higher categorical structures.

• 📦 Core Structures: Objects, morphisms, categories with full composition laws
• 🔄 Functors: Map between categories while preserving structure
• 🌀 Natural Transformations: Morphisms between functors with naturality checking
• 2️⃣ Higher Categories: 2-categories, ∞-categories, simplicial sets
• 🧮 Advanced Constructions: Monoidal, braided, enriched categories, and toposes
• 🔧 Algebraic Structures: Operads and algebras
• 📊 Diagram Tools: Automated commutativity checking
• 🏗️ Standard Categories: Pre-built categories (terminal, initial, discrete, etc.)

HyperCat brings the mathematical elegance of category theory to Python, designed for:

• 📐 Research in pure and applied category theory
• 🧮 Computational modeling of categorical structures
• 🧠 Algebraic topology and homotopy theory
• 🤖 Categorical foundations for functional programming and type theory
• 🔬 Mathematical foundations for machine learning and AI systems

git clone https://github.com/Mircus/HyperCat.git

cd HyperCat
pip install -e .

• Python 3.8 or higher
• Dependencies: networkx, matplotlib, jupyter, pytest

import hypercat
from hypercat import Category, Object, Morphism, Functor

# Create a category
cat = Category("Sets")

# Add objects
A = Object("A")
B = Object("B")
cat.add_object(A)
cat.add_object(B)

# Add morphisms
f = Morphism("f", A, B)
cat.add_morphism(f)

# Compose morphisms
g = Morphism("g", B, A)
cat.add_morphism(g)
h = cat.compose(f, g)  # h: A → A

print(f"Composed morphism: {h}")

# Create another category
cat2 = Category("Groups")
cat2.add_object(A)
cat2.add_object(B)
cat2.add_morphism(f)

# Create a functor
F = Functor("F", cat, cat2)
F.map_object(A, A)
F.map_object(B, B)
F.map_morphism(f, f)

# Check functor laws
print(f"Preserves composition: {F.is_functor()}")

from hypercat import NaturalTransformation

# Create two functors F, G: cat → cat2
G = Functor("G", cat, cat2)
# ... set up G's mappings ...

# Create natural transformation η: F ⇒ G
eta = NaturalTransformation("η", F, G)
# ... set up components ...

# Verify naturality
print(f"Is natural: {eta.is_natural()}")

from hypercat import StandardCategories

# Pre-built categories
terminal = StandardCategories.terminal_category()  # Single object, single morphism
initial = StandardCategories.initial_category()    # Empty category
discrete = StandardCategories.discrete_category(["X", "Y", "Z"])  # Only identity morphisms

Start with our interactive Jupyter notebooks in the examples/ directory:

• 00_quickstart.ipynb: Complete introduction to HyperCat
• HyperCatDemo.ipynb: Advanced features demonstration
• hypercat_algebraic_topology_demo.ipynb: Algebraic topology applications
• hypercats.ipynb: Working with higher categories

hypercat/
├── core/           # Core classes: Category, Object, Morphism, Functor
├── categories/     # Specialized categories: Monoidal, Braided, Enriched, Topos
├── higher/         # Higher categories: 2-categories, ∞-categories
├── diagram/        # Diagram and commutativity checking tools
└── algebraic/      # Operads and algebraic structures

• Category: The fundamental structure with objects and morphisms
• Functor: Structure-preserving mappings between categories
• NaturalTransformation: Morphisms between functors
• TwoCategory: Categories with 2-cells (morphisms between morphisms)
• MonoidalCategory: Categories with tensor products
• Topos: Categories with all finite limits and power objects

Run the test suite:

pytest

See ROADMAP.md for planned features:

• Universal properties and (co)limits
• Kan extensions
• Model categories
• Higher topos theory
• Categorical logic
• Integration with proof assistants

Check out our ROADMAP.md for upcoming features, including:

• (Co)limits and universal constructions
• Functorial logic and transformation categories

We welcome contributions! See CONTRIBUTING.md for how to get started.

"The fabric of higher structures"