Koszul duality of the category of trees and bar construction for operads

TL;DR

This paper establishes that the category of trees is Koszul, linking it to operad bar constructions and providing new insights into operads up to homotopy through categorical and homological perspectives.

Contribution

It proves the Koszulity of the category of trees and relates it to operad bar constructions and homotopy operads, offering a new categorical framework.

Findings

Category of trees is Koszul

Reduced bar construction interpreted as Koszul complex

Operads up to homotopy as functors from minimal resolution

Abstract

In this paper we study a category of trees TI and prove that it is a Koszul category. Consequences are the interpretation of the reduced bar construction of operads of Ginzburg and Kapranov as the Koszul complex of this category, and the interpretation of operads up to homotopy as a functor from the minimal resolution of TI to the category of graded vector spaces. We compare also three different bar constructions of operads. Two of them have already been compared by Shnider-Von Osdol and Fresse.