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

TL;DR

This paper introduces new functors connecting nonsymmetric operads and associative algebras, explores their Koszul duality relationship, and applies these concepts to provide a homotopical algebra proof of the Lagrange inversion formula.

Contribution

It defines and analyzes enveloping operad functors as adjoints to a forgetful functor, revealing their Koszul duality and applying them to prove the Lagrange inversion formula homotopically.

Findings

Enveloping operad functors are related by Koszul duality.

Limitations of standard Koszul duality tools are demonstrated.

Homotopical algebra provides a proof of the Lagrange inversion formula.

Abstract

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