Swiss-cheese action on the totalization of action-operads

Abstract

We prove that, if a pair of semicosimplicial spaces $\left(X_c^{\bullet };X_o^{\bullet }\right)$ arises from a colored operad, then the semitotalization $\text{ sTot}\left(X_o^{\bullet }\right)$ has the homotopy type of a relative double loop space and the pair $\left(\text{ sTot}\left(X_c^{\bullet }\right);\text{ sTot}\left(X_o^{\bullet }\right)\right)$ is weakly equivalent to an explicit algebra over the two dimensional Swiss-cheese operad ${\mathcal{S}\mathcal{C}}_2$.