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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 19th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIf we take the Mac Lane/Lawvere-Rosebrugh/[Becker-Gottlieb](https://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf) 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](http://www.cornell.edu/video/?videoID=2099).

   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.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeJul 19th 2012
   • PermaLink
   Author: zskoda
   Format: MarkdownItexHow 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](http://www.ams.org/mathscinet-getitem?mr=2200850), [doi](http://dx.doi.org/10.1016/j.aim.2005.11.004)

   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
3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 19th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexInteresting: >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.

   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.