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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E-theory categories for locally compact Hausdorff spaces and finite-open locales are equivalent to E-valued sheaves and cosheaves respectively.
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Bredon sheaf cohomology
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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$E$-theory of $X$-$C^{*}$-algebras and functor formalisms
E-theory categories for locally compact Hausdorff spaces and finite-open locales are equivalent to E-valued sheaves and cosheaves respectively.