# Delooping derived mapping spaces of bimodules over an operad

## TL;DR

This paper constructs a cofibrant replacement of an operad in bimodules, showing that the derived mapping space has a structure related to the little cubes operad, and establishes an explicit weak equivalence to a loop space.

## Contribution

It introduces a cofibrant replacement of operads in bimodules and relates the derived mapping space to a $ ext{C}_1$-algebra, providing explicit weak equivalences.

## Key findings

- The derived mapping space of bimodules is an algebra over the little cubes operad.
- An explicit weak equivalence from the loop space of operad maps to the bimodule mapping space is constructed.
- The approach clarifies the homotopical structure of bimodule mapping spaces over operads.

## 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 $Bimod_{O}^{h}(O;O')$.