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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 19th 2012

    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 (,1)(\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.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJul 19th 2012

    How about comparing to

    • W. G. Dwyer, J. P. C. Greenlees, S. Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006), no. 2, 357–402, MR2006k:55017, doi
    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 19th 2012


    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.