Condensation of the operad for multiplicative hyperoperads

Abstract

We construct a map of operads from an $E_2$-operad to the condensation of the operad for multiplicative hyperoperads. We deduce from it the existence of an $E_2$-action on the homotopy limit of the underlying functor of a multiplicative hyperoperad. This result is the higher dimensional analogue of a result due to Batanin and Berger implying Deligne's conjecture.