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

Koszul duality for monoids and the operad of enriched rooted trees