Free loop spaces and the Cauchy--Frobenius Lemma

2025/07/02 by Kock, Joachim, Mikhail, Thomas Jan
#18N60 #20-XX #55P35 #58D19 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2507.01637

Abstract

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.

Citations

• Groupoid Cardinality and Random Permutations
• Homotopy linear algebra
• Loop Groupoids, Gerbes, and Twisted Sectors on Orbifolds
• From Finite Sets to Feynman Diagrams

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