Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: supr-con
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-ex
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-ph
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The categories of this paper are: cs
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: nlin