Koszul duality for monoids and the operad of enriched rooted trees

TL;DR

This paper develops a Koszul duality theory for monoids in species and applies it to operads of enriched rooted trees, establishing conditions under which these operads are Koszul, thus linking algebraic and operadic dualities.

Contribution

It introduces Koszul duality for monoids in species and proves that the operad of enriched rooted trees is Koszul if and only if the underlying monoid is Koszul, extending existing results.

Findings

Operad of enriched rooted trees is Koszul iff the monoid is Koszul.

Establishes a connection between Koszul duality for algebras and operads.

Provides a broad class of Koszul operads extending previous work.

Abstract

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.