ComCH

ComCH is a specialized computer algebra system for the study of commutativity up to coherent homotopies. It provides effective constructions suitable for concrete computations in algebraic topology.

See the documentation for the API reference and guided notebooks.

Motivation

Commutativity up to coherent homotopies originates in algebraic topology and has found modern uses in topological data analysis, motion planning, condensed matter physics, and other areas. Many of the surrounding mathematical ideas are defined non-constructively; ComCH helps bridge the gap between those theoretical concepts and concrete applications.

Mathematical overview

Following the pioneering work of Steenrod, Cartan, Adem, Serre, Araki-Kudo, Dyer-Lashof, Stasheff, Boardman-Vogt, May, and others, the modern framework for commutativity up to coherent homotopies is provided by operads and PROPs. $E_n$-operads parameterize different levels of homotopical commutativity. ComCH focuses on chain complexes and implements two models of the $E_\infty$-operad equipped with filtrations by $E_n$-operads: the McClure-Smith surjection operad [McS] and the Berger-Fresse Barratt-Eccles operad [BF]. It also provides effective constructions of cochain-level May-Steenrod operations [KMM], even-prime Adem coboundaries, and Cartan coboundaries at both even and odd primes [Med20], [CMM].

Installation

ComCH is written in pure Python and has no runtime dependencies:

python3 -m pip install comch

Jupyter notebooks

To run the notebooks locally, install the notebook extra and register its Python kernel:

python3 -m pip install "comch[notebooks]"
python3 -m ipykernel install --user --name comch --display-name "Python (comch)"
jupyter lab

The notebook files live in the notebooks directory of the source repository; they are not installed into site-packages by the notebook extra. From a repository checkout, open an example notebook and select the Python (comch) kernel. The notebooks can also be run directly in Binder using the badge at the top of this page.

Development

Install the development extra from a repository checkout:

python3 -m pip install -e ".[dev]"

References

[McS]: J. McClure, and J. Smith. "Multivariable cochain operations and little n-cubes." Journal of the American Mathematical Society 16.3 (2003): 681-704. DOI

[BF]: C. Berger, and B. Fresse. "Combinatorial operad actions on cochains." Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 137. No. 1. Cambridge University Press, 2004. DOI

[KMM]: R. M. Kaufmann and A. M. Medina-Mardones. "Cochain level May-Steenrod operations." Forum Mathematicum 33 (2021), no. 6, 1507-1526. DOI

[Med20]: A. M. Medina-Mardones. "An effective proof of the Cartan formula: the even prime." Journal of Pure and Applied Algebra 224 (2020), no. 12, 106444. DOI

[CMM]: F. Cantero-Morán and A. M. Medina-Mardones. "An effective proof of the Cartan formula: odd primes." Homology, Homotopy and Applications 27 (2025), no. 1, 207-234. DOI