A computer algebra system for the study of commutativity up-to-coherent homotopies

Abstract: The Python package ComCH is a lightweight specialized computer algebra system that provides models for well known objects, the surjection and Barratt-Eccles operads, parameterizing the product structure of algebras that are commutative in a derived sense. The primary examples of such algebras treated by ComCH are the cochain complexes of spaces, for which it provides effective constructions of Steenrod cohomology operations at all prime.

• The paper introduces ComCH, a novel computer algebra system that models derived commutativity using operadic frameworks.
• It computes Steenrod operations across all primes, addressing challenges with odd primes through chain-level invariants.
• The system employs both surjection and Barratt-Eccles operads, setting the stage for advanced homotopical algebra research.

A Computer Algebra System for the Study of Commutativity Up-to-Coherent Homotopies

Introduction

The study of algebraic structures within mathematics often revolves around the concept of commutativity. Historically, this was believed to be a universal property across number systems until the discovery of quaternions by Hamilton, which demonstrated that non-commutative algebraic structures could exist. This has profound implications akin to those of non-Euclidean geometries. As algebra evolved, the development of topology and homotopy theory necessitated a reevaluation of commutativity, leading to the exploration of systems that correct non-commutativity via homotopies. The paper "A Computer Algebra System for the Study of Commutativity Up-to-Coherent Homotopies" (2102.07670) introduces ComCH, a specialized computer algebra system designed to model these complex algebraic structures. Specifically, it targets algebras commutative in a derived sense, providing effective computations for Steenrod cohomology operations, thus serving as an essential tool in research areas like topological data analysis and condensed matter physics.

Main Contributions

Computer Algebra System: ComCH

ComCH is a Python-based lightweight system that models the surjection and Barratt-Eccles operads to describe product structures in derived commutative algebras, notably cochain complexes. These operads are pivotal in modern algebraic topology, offering frameworks for structuring homological and homotopical data. By leveraging operads filtered by $E_n$-operads, ComCH facilitates the computation of homotopically enriched product structures.

Steenrod Operations and Effective Computations

The primary novelty of ComCH lies in its computation of Steenrod operations across all primes. Traditionally, describing such operations, particularly at odd primes, lacked implementation in algebraic software, which ComCH now addresses. This capability is crucial for researchers requiring exact chain-level representations when working with spaces depicted as simplicial or cubical sets.

Methodology

Operadic Framework: The system utilizes two operadic models, the surjection operad by McClure-Smith and the Barratt-Eccles operad by Berger-Fresse, each equipped with distinct sign conventions. These operads are implemented to parameterize different levels of derived commutativity and to resolve $R$ as an $R[_r]$-module.

Symmetric Groups and Free Modules: ComCH models algebraic structures using classes that represent elements in symmetric group rings and free modules, enabling complex algebraic manipulations and computations.

Steenrod Structures: The system introduces Steenrod-Adem structures for the homological study of algebras over $E_n$-operads, producing chain-level invariants instrumental in various mathematical and physical contexts.

Implications and Future Work

The implementation of ComCH extends beyond its current capabilities of Steenrod operations. Future developments might include:

• Chain-Level Operations for $E_n$-Algebras: Examining secondary operations and their relations in homological structures, further enriching the algebraic framework.
• Permutahedral Structures: The exploration of permutahedral chains could enhance the modeling of double cobar constructions, a potential advancement for homotopical data structures.
• Extended Implementation of Cartan and Adem Relations: While current implementations focus on $p=2$, expanding these to odd primes could unify and simplify homological algebra computations.

Conclusion

The paper "A Computer Algebra System for the Study of Commutativity Up-to-Coherent Homotopies" (2102.07670) provides a significant contribution to the field of algebraic topology and homological algebra. By developing ComCH, the author delivers a robust computational framework for handling operadic structures, Steenrod operations, and homotopical data. These advancements are poised to aid researchers in both theoretical investigations and practical applications, offering a template for future extensions into unexplored algebraic territories.