Shuffle polygraphic resolutions for operads

TL;DR

This paper introduces shuffle polygraphs as a categorical framework for rewriting in shuffle operads, generalizing Gr"obner bases by removing monomial order constraints, and proves Koszulness for certain quadratic cases.

Contribution

It develops shuffle polygraphs as a new categorical model for operad rewriting, extending Gr"obner bases, and establishes Koszulness for quadratic convergent cases.

Findings

Shuffle polygraphs generalize Gr"obner bases without monomial order constraints.

Convergent shuffle polygraphs can be extended into resolutions.

Quadratic convergent shuffle polygraphs present Koszul operads.

Abstract

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 Gr\"obner 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 Gr\"obner bases approach by removing the constraint of a monomial order for the orientation of the rewriting rules. We define w-operads as internal w-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.