Bredon sheaf cohomology computes algebraic K-theory of equivariant sheaves and equivariant E-theory of function algebras on G-spaces while satisfying a strong uniqueness theorem via open descent and compact codescent.
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The Galois groupoid of G-spectra is equivalent to the étale fundamental groupoid of the Burnside ring of G.
Higher algebra notions are generalized to t-structured tensor triangulated ∞-categories, with analogues of Lazard's theorem, Cohn localizations, almost ring theory, étale rigidity, and a moduli characterization under projective rigidity.
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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.