Condensation of the operad for multiplicative hyperoperads

TL;DR

This paper constructs a new operad map linking an $E_2$-operad to hyperoperad structures, establishing an $E_2$-action on certain homotopy limits, extending known results related to Deligne's conjecture.

Contribution

It introduces a novel operad map from an $E_2$-operad to the condensation of hyperoperad operads, generalizing previous results to higher dimensions.

Findings

Established an $E_2$-action on the homotopy limit of a multiplicative hyperoperad

Constructed a map from an $E_2$-operad to hyperoperad condensations

Extended Batanin and Berger's result to higher-dimensional contexts

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.