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Journal Articles Journal of the London Mathematical Society Year : 2023

Shuffle polygraphic resolutions for operads

Philippe Malbos
Isaac Ren
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Shuffle operads were introduced to make explicit the actions of symmetric groups on symmetric operads. 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 general framework 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-operads as internal ω-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.
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Dates and versions

hal-03161522 , version 1 (07-03-2021)
hal-03161522 , version 2 (08-07-2023)



Philippe Malbos, Isaac Ren. Shuffle polygraphic resolutions for operads. Journal of the London Mathematical Society, 2023, 107 (1), pp.61-122. ⟨10.1112/jlms.12681⟩. ⟨hal-03161522v2⟩
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