Theoretical analysis demonstrates a homotopy equivalence of infinity-groupoids via quotient free loop spaces, indicating a higher-categorical generalization of the Cauchy-Frobenius lemma.
We upgrade the Cauchy--Frobenius Lemma (`Burnside's Lemma') to a homotopy equivalence of ∞-groupoids, essentially given by double counting/Fubini in the free loop space of the quotient.
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Kock et al. (2026) studied this question.
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