Relative Recognition Principle

Abstract

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 $({\Omega }_{\text{rel}}^N(\iota :B\hookrightarrow Y),{\Omega }^N(Y))$ 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 $2<N\le \infty$, in the form of an equivalence of homotopy categories.

40 pages, 1 figure