Shuffle polygraphic resolutions for operads

• Published in: Journal of the London Mathematical Society 2023-01, Vol.107 (1), p.61-122
• Main Authors: Malbos, Philippe, Ren, Isaac
• Format: Article
• Language: English
• Subjects: Category Theory; K-Theory and Homology; Mathematics; Rings and Algebras
• ISSN: 0024-6107; 1469-7750; 1469-7750
• DOI: 10.1112/jlms.12681

Shuffle operads were introduced to forget the symmetric group actions on symmetric operads while preserving all possible operadic compositions. Rewriting methods were then applied to symmetric operads via shuffle operads: in particular, a notion of Gröbner basis was introduced for shuffle operads with respect to a total order on tree monomials. In this article, we introduce the structure of shuffle polygraphs as a categorical model for rewriting in shuffle operads, which generalizes the Gröbner bases approach by removing the constraint of a monomial order for the orientation of the rewriting rules. We define ω$\omega$‐operads as internal ω$\omega$‐categories in the category of shuffle operads. We show how to extend a convergent shuffle polygraph into a shuffle polygraphic resolution generated by the overlapping branchings of the original polygraph. Finally, we prove that a shuffle operad presented by a quadratic convergent shuffle polygraph is Koszul.