Relative recognition principle

Abstract

We prove the recognition principle for relative $N$–loop pairs of spaces for $3\le N\le \infty$. If $3\le N<\infty$, this states that a pair of spaces homotopy equivalent to CW–complexes $\left(X_c,X_o\right)$ is homotopy equivalent to $\left(Y^{{\mathbb{𝕊}}^N},HFib{\left(\iota \right)}^{{\mathbb{𝕊}}^{N-1}}\right)$ for a functorially determined relative space $\iota :B\to Y$ if and only if $\left(X_c,X_o\right)$ is a grouplike ${\overline{\mathcal{𝒮}\mathcal{𝒞}}}_N$–space, where ${\overline{\mathcal{𝒮}\mathcal{𝒞}}}_N$ is any cofibrant resolution of the Swiss-cheese relative operad ${\mathcal{𝒮}\mathcal{𝒞}}_N$. The relative recognition principle for relative $\infty$–loop pairs of spaces states that a pair of spaces $\left(X_c,X_o\right)$ homotopy equivalent to CW–complexes is homotopy equivalent to $\left(Y_0,HFib\left({\iota }_0\right)\right)$ for a functorially determined relative spectrum ${\iota }_{\bullet }:B_{\bullet }\nearrow Y_{\bullet +1}$ if and only if $\left(X_c,X_o\right)$ is a grouplike ${\mathcal{ℰ}}^{\to }$–algebra, where ${\mathcal{ℰ}}^{\to }$ is a contractible cofibrant relative operad or equivalently a cofibrant resolution of the terminal relative operad ${Com}^{\to }$ of continuous homomorphisms of commutative monoids. These principles are proved as equivalences of homotopy categories.