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 morphismsStart 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 morphismsFunctor: Structure-preserving mappings between categoriesNaturalTransformation: Morphisms between functorsTwoCategory: Categories with 2-cells (morphisms between morphisms)MonoidalCategory: Categories with tensor productsTopos: Categories with all finite limits and power objects
Run the test suite:
pytestSee 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"