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

Muriel Livernet (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications

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.

Mots clés: operads; bar construction; monad; Koszul duality

Domaines: Anneaux et algèbres [math.RA]; Catégories et ensembles [math.CT]

Muriel Livernet. Koszul duality of the category of trees and bar construction for operads. 2011. ⟨hal-00566707⟩