Simplicial Tensor Operations

Simplicial operations on matrices and hypermatrices.

Mathematical formalities

This project defines simplicial operations matrices and hypermatrices (tensors), including face maps (d_i), degeneracies (s_i), and boundary operators, as well mathematical computer experiments for the diagonal simplicial tensor module X(s; A), where s=(n_1,...,n_k) is the shape of a tensor of order k.

The operations defined here satisfy the standard simplicial identities.

$$ \begin{aligned} d_i d_j &= d_{j-1} d_i, && \text{if } i < j; \\ s_i s_j &= s_j s_{i-1}, && \text{if } i > j; \\ d_i s_j &= \begin{cases} s_{j-1} d_i, & \text{if } i < j; \\ 1, & \text{if } i \in {j, j+1}; \\ s_j d_{i-1}, & \text{if } i > j+1. \end{cases} \end{aligned} $$

To work with these files on your own machine:

1. Clone or update the repository (git clone the first time, otherwise git pull).

2. Change into the Operations-Tensorielles-Simpliciales directory.

3. Install the package in editable mode:

   pip install -e .

You can then import modules via import simplicial_tensors.tensor_ops.

Examples

The interactive demonstrations that used to live inside the library modules now reside in the examples/ folder. After installing the project in editable mode (pip install -e .), run any example like so:

python examples/tensor_ops.py

Each script imports the corresponding module under simplicial_tensors and calls its main() entry point.