Relative Recognition Principle

In this paper the relative recognition principle will be proved. It states that a pair of spaces $(X_o,X_c)$ is weakly equivalent to $({\mathrm{\Omega }}_{\text{rel}}^N(\iota :B\hookrightarrow Y),{\mathrm{\Omega }}^N(Y))$ if and only if $(X_o,X_c)$ are grouplike $\overline{{\mathcal{S}\mathcal{C}}^N}$-spaces, where $\overline{{\mathcal{S}\mathcal{C}}^N}$ is any cofibrant resolution of the Swiss-cheese 2-operad ${\mathcal{S}\mathcal{C}}^N$. This principle will be proved for connected $\overline{{\mathcal{S}\mathcal{C}}^N}$-spaces, and also for grouplike $\overline{{\mathcal{S}\mathcal{C}}^N}$-spaces for $2<N\le \mathrm{\infty }$, in the form of an equivalence of homotopy categories.