Delooping derived mapping spaces of bimodules over an operad

Abstract: From a map of operads $\eta : O\rightarrow O'$, we introduce a cofibrant replacement of the operad $O$ in the category of bimodules over itself such that the corresponding model of the derived mapping space of bimodules $Bimod_{O}^{h}(O;O')$ is an algebra over the one dimensional little cubes operad $\mathcal{C}{1}$. In the present work, we also build an explicit weak equivalence of $\mathcal{C}{1}$-algebras from the loop space $\Omega Operad^{h}(O;O')$ to $Bimod_{O}^{h}(O;O')$.