A Quillen adjunction between algebras and operads, Koszul duality, and the Lagrange inversion formula

摘要

We define, for a somewhat standard forgetful functor from nonsymmetricoperads to weight graded associative algebras, two functorial "envelopingoperad" functors, the right inverse and the left adjoint of the forgetfulfunctor. Those functors turn out to be related by operadic Koszul duality, andthat relationship can be utilised to provide examples showing limitations oftwo standard tools of the Koszul duality theory. We also apply these functorsto get a homotopical algebra proof of the Lagrange inversion formula.