Triple delooping for multiplicative hyperoperads

TL;DR

This paper extends the double delooping results to higher iterations of the Baez-Dolan plus construction, introducing new bimodule notions and conditions for triple delooping in the context of hyperoperads.

Contribution

It generalizes the double delooping to multiple iterations of the Baez-Dolan construction and introduces $(m,n)$-bimodules extending previous bimodule concepts.

Findings

Double delooping exists for the introduced bimodules.

Triple delooping achieved under a new reduceness condition.

Extends delooping techniques to hyperoperads and higher iterations.

Abstract

Using techniques developed by Batanin and the first author, we extend the Turchin/Dwyer-Hess double delooping result to further iterations of the Baez-Dolan plus construction. For $0 \leq m \leq n$, we introduce a notion of $(m,n)$-bimodules which extends the notions of bimodules and infinitesimal bimodules over the terminal non-symmetric operad. We show that a double delooping always exists for these bimodules. For the triple iteration of the Baez-Dolan construction starting from the initial $1$-coloured operad, we provide a further reduceness condition to have a third delooping.