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 $(\Omega^N_\text{rel}(\iota:B\hookrightarrow Y),\Omega^N(Y))$ if and only if $(X_o,X_c)$ are grouplike $\overline{\mathcal{SC}^N}$-spaces, where $\overline{\mathcal{SC}^N}$ is any cofibrant resolution of the Swiss-cheese 2-operad $\mathcal{SC}^N$. This principle will be proved for connected $\overline{\mathcal{SC}^N}$-spaces, and also for grouplike $\overline{\mathcal{SC}^N}$-spaces for $2<N\leq\infty$, in the form of an equivalence of homotopy categories.