Koszul duality for monoids and the operad of enriched rooted trees

We introduce here the notion of Koszul duality for monoids in the monoidal category of species with respect to the ordinary product. To each Koszul monoid we associate a class of Koszul algebras in the sense of Priddy, by taking the corresponding analytic functor. The operad $\mathscr{A}_M$ of rooted trees enriched with a monoid $M$ was introduced by the author many years ago. One special case of that is the operad of ordinary rooted trees, called in the recent literature the permutative non associative operad. We prove here that $\mathscr{A}_M$ is Koszul if and only if the corresponding monoid $M$ is Koszul. In this way we obtain a wide family of Koszul operads, extending a recent result of Chapoton and Livernet, and providing an interesting link between Koszul duality for associative algebras and Koszul duality for operads.

💡 Research Summary

The paper develops a new notion of Koszul duality for monoids in the monoidal category of combinatorial species equipped with the ordinary (Cauchy) product. A species is a functor F from the groupoid of finite sets and bijections to sets; the ordinary product (denoted “·”) of two species F and G is defined by (F·G)