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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 15th 2013
   • (edited Nov 15th 2013)
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   Author: Urs
   Format: MarkdownItexI noticed that some old entries were requesting a keyword link for _[[brave new algebra]]_, so I created it and filled in a default-paragraph. Please feel invited to expand. In that context I have a question: the [dual generlized Steenrod algebras](http://ncatlab.org/nlab/show/Steenrod+algebra#HopfAlgebraStructure) have been called "brave new Hopf algebroids" in articles including * [[Andrew Baker]], _Brave new Hopf algebroids_ ([pdf](http://www.maths.gla.ac.uk/~ajb/dvi-ps/brave-ha.pdf)) * [[Andrew Baker]] and Alain Jeanneret, _Brave new Hopf algebroids and extensions of $MU$-algebras_, Homology Homotopy Appl. Volume 4, Number 1 (2002), 163-173. ([Euclid](http://projecteuclid.org/euclid.hha/1139840059)) * [[Mark Hovey]], _Homotopy theory of comodules over a Hopf algebroid_ ([arXiv:math/0301229](http://arxiv.org/abs/math/0301229)) But the Hopf algebroids considered in these articles are ordinary Hopf algebroids, they are given not by Hopf $\infty$-algebras $(E, E \wedge E)$ but by their homotopy groups $(E_\bullet, E_\bullet(E))$, unless I am missing something. So at least without further discussion, calling $(E_\bullet, E_\bullet(E))$ "brave new" is a bit of a stretch. The brave new thing would be $(E, E \wedge E)$ (if indeed it is a "Hopf $\infty$-algebroid"). Can anyone say more about this? I can't seem to find any source talking about this. The canonical guess of googling for "derived Hopf algebroid" doesn't show relevant results.

   I noticed that some old entries were requesting a keyword link for brave new algebra, so I created it and filled in a default-paragraph. Please feel invited to expand.

   In that context I have a question: the dual generlized Steenrod algebras have been called “brave new Hopf algebroids” in articles including

   • Andrew Baker, Brave new Hopf algebroids (pdf)

   • Andrew Baker and Alain Jeanneret, Brave new Hopf algebroids and extensions of $MU$-algebras, Homology Homotopy Appl. Volume 4, Number 1 (2002), 163-173. (Euclid)

   • Mark Hovey, Homotopy theory of comodules over a Hopf algebroid (arXiv:math/0301229)

   But the Hopf algebroids considered in these articles are ordinary Hopf algebroids, they are given not by Hopf $\infty$-algebras $(E, E \wedge E)$ but by their homotopy groups $(E_\bullet, E_\bullet(E))$, unless I am missing something.

   So at least without further discussion, calling $(E_\bullet, E_\bullet(E))$ “brave new” is a bit of a stretch. The brave new thing would be $(E, E \wedge E)$ (if indeed it is a “Hopf $\infty$-algebroid”).

   Can anyone say more about this? I can’t seem to find any source talking about this. The canonical guess of googling for “derived Hopf algebroid” doesn’t show relevant results.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeNov 15th 2013
   • (edited Nov 15th 2013)
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   Author: zskoda
   Format: MarkdownItex> in these articles are ordinary Hopf algebroids ...and commutative. I think we should really split the entry on commutative Hopf algebroids. This way the balance in the entry is misleading to the generalal theory where the noncommutative base makes theory building necessary in quite unexpected way in comparison to just Gelfand-dualizing usual groupoid theory (the monoid and comonoid structure are in different monoidal categories!). As far as higher categorical versions, the most nontrivial works are those of Baues in which he developed bicategorical analogue of Steenrod algebra, the secondary Steenrod algebra, which encodes all information on __secondary__ cohomological operations.

   in these articles are ordinary Hopf algebroids

   …and commutative. I think we should really split the entry on commutative Hopf algebroids. This way the balance in the entry is misleading to the generalal theory where the noncommutative base makes theory building necessary in quite unexpected way in comparison to just Gelfand-dualizing usual groupoid theory (the monoid and comonoid structure are in different monoidal categories!).

   As far as higher categorical versions, the most nontrivial works are those of Baues in which he developed bicategorical analogue of Steenrod algebra, the secondary Steenrod algebra, which encodes all information on secondary cohomological operations.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 15th 2013
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   Author: Urs
   Format: MarkdownItexThanks, I should look into this, then.

   Thanks, I should look into this, then.