Shuffle polygraphic resolutions for operads

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