On a notion of homotopy Segal $ E_\infty $-Hopf cooperad

Abstract: We define a notion of homotopy Segal cooperad in the category of $ E_\infty $-algebras. This model of Segal cooperad that we define in the paper, which we call homotopy Segal $ E_\infty $-Hopf cooperad, covers examples given by the cochain complex of topological operads and provides a framework for the study of the homotopy of such objects. In a first step, we consider a category of Segal $ E_\infty $-Hopf cooperads, which consists of collections of $ E_\infty $-algebras indexed by trees and equipped with coproduct operators, corresponding to tree morphisms, together with facet operators, corresponding to subtree inclusions. The coproduct operators model coproducts of operations inside a tree. The facet operators are assumed to satisfy a Segal condition. The homotopy Segal cooperads that we aim to define are formed by integrating homotopies in the composition schemes of the coproduct operators. For this purpose, we replace the functorial structure that governs the composition of the coproduct operators by the structure of a homotopy functor which we shape on a cubical enrichment of the category of $ E_\infty $-algebras. We prove that every homotopy Segal $ E_\infty $-Hopf cooperad in our sense is weakly-equivalent to a strict Segal $ E_\infty $-Hopf cooperad. We also define a notion of homotopy morphism of homotopy Segal $ E_\infty $-Hopf cooperads. We prove that every homotopy Segal $ E_\infty $-Hopf cooperad admits a cobar construction and that every homotopy morphism of homotopy Segal $ E_\infty $-Hopf cooperads induces a morphism on this cobar construction, so that our approach provides a lifting to the context of $ E_\infty $-algebras of classical homotopy cooperad structures that are modeled on the bar duality of operads when we work in a category of differential graded modules.