A∞-actions and recognition of relative loop spaces

We show that relative loop spaces are recognized by A∞A∞-actions. A certain version of the 2-sided bar construction is used to prove such recognition theorem. The operad Act∞Act∞ of A∞A∞-actions is presented in terms of the Boardman–Vogt resolution of the operad ActAct. We exhibit an operad homotopy equivalence between such resolution and the 1-dimensional Swiss-cheese operad SC1SC1.