Delooping derived mapping spaces of bimodules over an operad

To any topological operad O, we introduce a cofibrant replacement in the category of bimodules over itself such that for every map \(\eta :O\rightarrow O'\) of operads, the corresponding model \({\textit{Bimod}}_{O}^{h}(O\,;\,O')\) of derived mapping space of bimodules is an algebra over the one dimensional little cubes operad \(\mathcal {C}_{1}\). We also build an explicit weak equivalence of \(\mathcal {C}_{1}\)-algebras from the loop space \(\Omega {\textit{Operad}}^{h}(O\,;\,O')\) to \({\textit{Bimod}}_{O}^{h}(O\,;\,O')\).