Resolutions of operads via Koszul (bi)algebras

Abstract

We introduce a construction that produces from each bialgebra H an operad \(\mathsf {Ass}_H\) controlling associative algebras in the monoidal category of H-modules or, briefly, H-algebras. When the underlying algebra of this bialgebra is Koszul, we give explicit formulas for the minimal model of this operad depending only on the coproduct of H and the Koszul model of H. This operad is seldom quadratic—and hence does not fall within the reach of Koszul duality theory—so our work provides a new rich family of examples where an explicit minimal model of an operad can be obtained. As an application, we observe that if we take H to be the mod-2 Steenrod algebra \({\mathscr {A}}\), then this notion of an associative H-algebra coincides with the usual notion of an \(\mathscr {A}\)-algebra considered by homotopy theorists. This makes available to us an operad \(\mathsf {Ass}_{{\mathscr {A}}}\) along with its minimal model that controls the category of associative \({\mathscr {A}}\)-algebras, and the notion of strong homotopy associative \({\mathscr {A}}\)-algebras.