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If we take the Mac Lane/Lawvere-Rosebrugh/Becker-Gottlieb division of duality into
axiomatic/formal/Eckmann-Hilton: arrow reversal
functional/concrete/strong: dualizing object, pairing,…
We see that the formal approach to dualizing cohomology to homotopy can withstand the move from ordinary (abelian) cohomology to the fanciest forms in the shape of arrows in and out in a $(\infty, 1)$-topos.
What then of the concrete dual of cohomology as homology? Do we get a concrete dual of the most general forms of cohomology? If Becker and Gottleib can provide the
unification of Poincaré, Alexander, Lefschetz, Spanier-Whitehead, homology-cohomology duality,
is there a similar unified notion in the generalized nonabelian setting?
Maybe I should listen to Lurie on The Siegel Mass Formula, Tamagawa Numbers, and Nonabelian Poincaré Duality.
How about comparing to
Interesting:
Maybe the most important contribution of this paper is the conceptual framework, which allows us to view all of the following dualities
• Poincaré duality for manifolds
• Gorenstein duality for commutative rings
• Benson-Carlson duality for cohomology rings of finite groups
• Poincaré duality for groups
• Gross-Hopkins duality in chromatic stable homotopy theory
as examples of a single phenomenon.
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