The Galois groupoid of G-spectra is equivalent to the étale fundamental groupoid of the Burnside ring of G.
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Topological symmetric homology equals the free E∞-algebra on an E1-algebra; topological braid homology equals the free E2-algebra on an E1-algebra, via identification of the E2-monoidal envelope with the braided crossed simplicial group.
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The Galois theory of $G$-spectra and the Burnside ring
The Galois groupoid of G-spectra is equivalent to the étale fundamental groupoid of the Burnside ring of G.
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Topological symmetric and braid homologies
Topological symmetric homology equals the free E∞-algebra on an E1-algebra; topological braid homology equals the free E2-algebra on an E1-algebra, via identification of the E2-monoidal envelope with the braided crossed simplicial group.