ComCH is a Python 3 package for the study of commutativity up-to-coherent-homotopies. Its documentation is located here.
Commutativity up-to-coherent-homotopies is a concept originated in algebraic topology. It has found modern uses in topological data analysis, motion planning, condensed matter physics and several other areas. An important challenge for the application of the mathematical ideas surrounding this concept, which are often defined non-constructively, is to describe them in effective terms suitable for concrete computations. This package, a specialized computer algebra system, serves to bridge this gap between theoretical concepts and concrete applications.
Following the pioneering work of Steenrod, Cartan, Adem, Stashef, Boardman-Vogt, May, Dyer-Lashof and others, today we understand the correct framework for the study of commutativity up-to-coherent-homotopies as the one provided by operads and PROPs. In particular,
This package is written in Python 3 and has no dependencies. It can be installed from a terminal entering:
python3 -m pip install comch
[McS]: J. McClure, and J. Smith. "Multivariable cochain operations and little n-cubes." Journal of the American Mathematical Society 16.3 (2003): 681-704.
[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.
[KMM]: Kaufmann, R. M., & Medina-Mardones, A. M. (2020). Chain level Steenrod operations. arXiv preprint arXiv:2010.02571.