Swiss-cheese action on the totalization of operads under the monoid actions actions operad

摘要

We prove that if a pair of semi-cosimplicial spaces (X,Y) arise from acoloured operad then the semi-totalization sTot(Y) has the homotopy type of arelative double loop space and the pair (sTot(X),sTot(Y)) is weakly equivalentto an explicit algebra over the two dimensional Swiss-cheese operad.