Using six functor formalisms, the authors prove that hypercomplete locally compact ANR homology manifolds are cohomologically smooth, that compact ANR homology manifolds are Poincaré duality complexes with Spivak tangent fibration as dualizing sheaf, introduce homotopy manifolds, and show that homot
The sheaves-spectrum adjunction
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Sheaf categories are the unique six functor formalisms on LCH spaces satisfying natural properties, implying equivalence for continuous formalisms.
The derived category of a locally compact Hausdorff space X is smooth if and only if X is discrete and finite.
An ind-Banach framework defines overconvergent and holomorphic power series rings to endow C-algebras with analytic structures, yielding an abstract GAGA-type comparison.
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Homology manifolds via six functor formalisms
Using six functor formalisms, the authors prove that hypercomplete locally compact ANR homology manifolds are cohomologically smooth, that compact ANR homology manifolds are Poincaré duality complexes with Spivak tangent fibration as dualizing sheaf, introduce homotopy manifolds, and show that homot
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A characterization of sheaves among six functor formalisms on $\mathrm{LCH}$
Sheaf categories are the unique six functor formalisms on LCH spaces satisfying natural properties, implying equivalence for continuous formalisms.
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The derived category of a locally compact space is rarely smooth
The derived category of a locally compact Hausdorff space X is smooth if and only if X is discrete and finite.
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Analytification for Complex Geometry Revisited
An ind-Banach framework defines overconvergent and holomorphic power series rings to endow C-algebras with analytic structures, yielding an abstract GAGA-type comparison.