# Cova Space
A comprehensive Rust library for computational topology and geometric analysis, providing rigorous implementations of topological spaces, simplicial complexes, homology computation, and topological data analysis.
[](https://crates.io/crates/cova-space)
[](https://docs.rs/cova-space)
[](https://www.gnu.org/licenses/agpl-3.0)
## Overview
Cova Space implements fundamental structures and algorithms from computational topology with a focus on mathematical rigor, type safety, and performance. The crate provides a comprehensive toolkit for topological computation, from basic set operations to advanced persistent homology and sheaf-theoretic constructions.
## Architecture
The library is organized around core topological concepts, building from foundational structures to sophisticated computational tools:
### Core Foundations
#### [`set`](src/set.rs)
Foundation layer providing collection abstractions and partially ordered sets (posets). Implements basic set operations, ordering relationships, and the mathematical framework for more complex topological structures.
#### [`definitions`](src/definitions.rs)
Fundamental trait hierarchy for mathematical spaces including topological spaces, metric spaces, normed spaces, and inner product spaces. Establishes the interface for geometric and topological operations with proper mathematical abstractions.
### Topological Complexes
#### [`complexes`](src/complexes/)
Comprehensive implementation of cell complexes including simplicial and cubical complexes. Provides generic complex containers, automatic face relation management, and efficient storage with ID-based lattice structures for computational topology applications.
**Submodules:**
- **`simplicial`**: Simplex definitions and simplicial complex operations
- **`cubical`**: Cube definitions and cubical complex operations
#### [`graph`](src/graph.rs)
Flexible graph data structures supporting both directed and undirected graphs with comprehensive operations for vertices, edges, and topological relationships. Designed for integration with complex and homological computations.
### Computational Topology
#### [`homology`](src/homology.rs)
Complete homology computation framework including chain complexes, boundary operators, and Betti number calculations. Implements formal chains with ring coefficients and supports homology computation over arbitrary fields for topological analysis.
#### [`sheaf`](src/sheaf.rs)
Advanced sheaf theory implementations providing categorical constructions over topological spaces. Includes restriction morphisms, global section verification, and coboundary operators for sophisticated topological data analysis.
### Topological Data Analysis
#### [`filtration`](src/filtration/)
Filtration frameworks for persistent homology including Vietoris-Rips constructions. Supports both serial and parallel computation of filtered complexes for analyzing multi-scale topological features in data.
#### [`cloud`](src/cloud.rs)
Point cloud analysis tools designed for topological data analysis applications. Provides the foundation for building filtered complexes from geometric data sets.
#### [`lattice`](src/lattice.rs)
Sophisticated lattice structures for efficient representation of partial orders and face relationships in complexes. Implements join/meet operations and provides the computational backbone for complex operations.
## Design Principles
- **Mathematical Rigor**: All implementations follow strict topological definitions and maintain structural invariants
- **Type Safety**: Leverages Rust's type system to encode topological properties and prevent invalid operations
- **Computational Efficiency**: Optimized data structures and algorithms for large-scale topological computations
- **Composability**: Modular design allows complex topological constructions from fundamental building blocks
## Usage
Add to your `Cargo.toml`:
```toml
[dependencies]
cova-space = "*"
```
The crate provides a comprehensive prelude for convenient importing:
```rust
use cova_space::prelude::*;
```
## Feature Highlights
- **Generic Complex Framework**: Unified interface for simplicial, cubical, and general cell complexes
- **Homology Computation**: Full chain complex machinery with boundary operators and Betti number calculation
- **Persistent Homology**: Filtration frameworks for multi-scale topological analysis
- **Sheaf Theory**: Advanced categorical constructions for topological data analysis
- **High Performance**: Efficient lattice-based storage and optional parallel computation support
## Optional Features
- **`parallel`**: Enables parallel computation for filtrations and large-scale operations using Rayon
## Mathematical Scope
The library covers essential areas of computational topology:
```
Set Theory & Posets
├── Topological Spaces (metric, normed, inner product)
├── Cell Complexes (simplicial, cubical, general)
├── Homological Algebra (chains, boundaries, homology)
├── Sheaf Theory (categorical constructions)
└── Topological Data Analysis (filtrations, persistence)
```
## Documentation
Complete API documentation is available on [docs.rs](https://docs.rs/cova-space).
## Contributing
Contributions are welcome! Please ensure mathematical correctness and include appropriate documentation for topological algorithms and data structures.
## License
This project is licensed under the AGPLv3 License - see the [LICENSE](../LICENSE) file for details.