Relative Recognition Principle

TL;DR

This paper proves a relative recognition principle linking pairs of spaces to certain loop space constructions, establishing an equivalence of homotopy categories for connected grouplike spaces.

Contribution

It establishes the relative recognition principle for pairs of spaces and proves the equivalence for connected and grouplike spaces using cofibrant resolutions of the Swiss-cheese 2-operad.

Findings

Proves the relative recognition principle for pairs of spaces.

Establishes homotopy category equivalence for connected grouplike spaces.

Uses cofibrant resolutions of the Swiss-cheese 2-operad.

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 $(\Omega^N_\text{rel}(\iota:B\hookrightarrow Y),\Omega^N(Y))$ 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.