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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJul 12th 2013
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   Author: Mike Shulman
   Format: MarkdownItexAdded to [[Hopf monad]] the Bruguières-Lack-Virelizier definition and some properties.

   Added to Hopf monad the Bruguières-Lack-Virelizier definition and some properties.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJul 12th 2013
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   Author: Mike Shulman
   Format: MarkdownItexNow added also the Mesablishvili-Wisbauer definition. There is a relationship between the two definitions described in the Mesablishvili-Wisbauer paper, but I can't really figure it out.

   Now added also the Mesablishvili-Wisbauer definition. There is a relationship between the two definitions described in the Mesablishvili-Wisbauer paper, but I can’t really figure it out.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeApr 26th 2023
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   Author: zskoda
   Format: MarkdownItexHopf (bi)monads are extensively studied in the book * [[Gabriella Böhm]], _Hopf algebras and their generalizations from a category theoretical point of view_, Lecture Notes in Math. __2226__, Springer 2018, [doi](htttps://doi.org/10.1007/978-3-319-98137-6) <a href="https://ncatlab.org/nlab/revision/diff/Hopf+monad/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hopf+monad/14">v14</a>, <a href="https://ncatlab.org/nlab/show/Hopf+monad">current</a>

   Hopf (bi)monads are extensively studied in the book

   • Gabriella Böhm, Hopf algebras and their generalizations from a category theoretical point of view, Lecture Notes in Math. 2226, Springer 2018, doi

   diff, v14, current

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeApr 26th 2023
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   Author: zskoda
   Format: MarkdownItexSomehow doi does not work directly but from <https://link.springer.com/book/10.1007/978-3-319-98137-6>

   Somehow doi does not work directly but from https://link.springer.com/book/10.1007/978-3-319-98137-6

5. • CommentRowNumber5.
   • CommentAuthorvarkor
   • CommentTimeMay 29th 2023
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   Author: varkor
   Format: MarkdownItexAdded a reference to Frobenius monads. <a href="https://ncatlab.org/nlab/revision/diff/Hopf+monad/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hopf+monad/15">v15</a>, <a href="https://ncatlab.org/nlab/show/Hopf+monad">current</a>

   Added a reference to Frobenius monads.

   diff, v15, current

6. • CommentRowNumber6.
   • CommentAuthorvarkor
   • CommentTimeMay 29th 2023
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   Author: varkor
   Format: MarkdownItexI'm not sure it's appropriate to redirect **bimonad** here, rather than having a disambiguation page, since other concepts (like [[Frobenius monads]]) could also be argued to deserve that name.

   I’m not sure it’s appropriate to redirect bimonad here, rather than having a disambiguation page, since other concepts (like Frobenius monads) could also be argued to deserve that name.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeSep 20th 2023
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   Author: Urs
   Format: MarkdownItexadded pointer to: * Masahito Hasegawa and Jean-Simon Pacaud Lemay, *Hopf Monads on Biproducts*, Theory and Applications of Categories **39** 28 (2023) 804-823 &lbrack;[tac:3928](http://www.tac.mta.ca/tac/volumes/39/28/39-28abs.html)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/Hopf+monad/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hopf+monad/16">v16</a>, <a href="https://ncatlab.org/nlab/show/Hopf+monad">current</a>

   added pointer to:

   • Masahito Hasegawa and Jean-Simon Pacaud Lemay, Hopf Monads on Biproducts, Theory and Applications of Categories 39 28 (2023) 804-823 [tac:3928]

   diff, v16, current

8. • CommentRowNumber8.
   • CommentAuthorperezl.alonso
   • CommentTimeNov 18th 2024
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   Author: perezl.alonso
   Format: MarkdownItexGiven an exact sequence of (finite-dim'l semisimple) Hopf algebras $A\to \tilde{H}\to H$, one has an exact sequence of fusion categories $\text{Comod}(A) \to \text{Comod} (\tilde{H}) \to \text{Comod} (H)$. Such exact sequences are classified by a Hopf monad on, in this case, $\text{Comod} (H)$, which here is just the composition $T=\text{Res} \circ \text{Ind}$ of the Induction functor $\text{Ind}$ followed by the Restriction functor $\text{Res}$ of comodules. In this case this is actually a Frobenius monad, as it has compatible lax and oplax structure. On the other hand, an exact sequence of Hopf algebras is determined by a weak action $H\otimes A\to A$, a cocycle $H\otimes H\to A$, a weak coaction $H\to H\otimes A$, and a dual cocycle $H\to A\otimes A$, satisfying some array of conditions. My question is, is there a straightforward way to state how these four ingredients construct the Frobenius monad $T=\text{Res} \circ \text{Ind}$ ? Or equivalently how starting from such a monad one can distil the weak (co)actions and (dual) cocycle?

   Given an exact sequence of (finite-dim’l semisimple) Hopf algebras , one has an exact sequence of fusion categories . Such exact sequences are classified by a Hopf monad on, in this case, , which here is just the composition of the Induction functor followed by the Restriction functor of comodules. In this case this is actually a Frobenius monad, as it has compatible lax and oplax structure. On the other hand, an exact sequence of Hopf algebras is determined by a weak action , a cocycle , a weak coaction , and a dual cocycle , satisfying some array of conditions.

   My question is, is there a straightforward way to state how these four ingredients construct the Frobenius monad ? Or equivalently how starting from such a monad one can distil the weak (co)actions and (dual) cocycle?

9. • CommentRowNumber9.
   • CommentAuthorperezl.alonso
   • CommentTimeJan 24th 2025
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   Author: perezl.alonso
   Format: MarkdownItexIn partial response to #8, at least for *abelian* extensions (Hopf algebra extensions of the form $k^{\Gamma}\to H\to kG$), this can be inferred from the $(G,\Gamma)$-crossed tensor structure on $Vec_{\Gamma}$ constructed in ([Natale 14](https://arxiv.org/abs/1405.6970)). I don't have time right now to contextualize this in the nLab article so I'll just leave this comment for the record.

   In partial response to #8, at least for abelian extensions (Hopf algebra extensions of the form ), this can be inferred from the -crossed tensor structure on constructed in (Natale 14). I don’t have time right now to contextualize this in the nLab article so I’ll just leave this comment for the record.