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1. • CommentRowNumber1.
   • CommentAuthorjoe.hannon
   • CommentTimeJun 2nd 2014
   • (edited Jun 2nd 2014)
   • PermaLink
   Author: joe.hannon
   Format: MarkdownItexThe article on [bimonoids](http://ncatlab.org/nlab/show/bimonoid) says that a bimonoid may be defined in a symmetric monoidal category, with some vague allusions that the definition still makes sense, with some difficulty, in a braided monoidal category or duoidal category (??). On the other hand, the article on [Hopf monoids]() says that the definition makes sense in any monoidal category. So which is it? For my part, I'm having trouble seeing how the definition even makes sense in a symmetric monoidal category. For example, you need the comultiplication to be a morphism of monoids, so in particular you need the codomain $M\otimes M$ of the comultiplication to be a monoid. There's an obvious multiplication $(M\otimes M)\otimes(M\otimes M)\to M\otimes M$, but it doesn't seem to satisfy the associativity axiom without inserting a bunch of monoidal symmetry swaps. This seems like a weaker form of associativity than is allowed by the monoid axioms, right? If $(M,\mu,\eta)$ is a monoid, then is $(M\otimes M,\mu\otimes\mu,\eta\otimes\eta)$ also a monoid?

   The article on bimonoids says that a bimonoid may be defined in a symmetric monoidal category, with some vague allusions that the definition still makes sense, with some difficulty, in a braided monoidal category or duoidal category (??). On the other hand, the article on Hopf monoids says that the definition makes sense in any monoidal category. So which is it?

   For my part, I’m having trouble seeing how the definition even makes sense in a symmetric monoidal category. For example, you need the comultiplication to be a morphism of monoids, so in particular you need the codomain $M\otimes M$ of the comultiplication to be a monoid. There’s an obvious multiplication $(M\otimes M)\otimes(M\otimes M)\to M\otimes M$, but it doesn’t seem to satisfy the associativity axiom without inserting a bunch of monoidal symmetry swaps. This seems like a weaker form of associativity than is allowed by the monoid axioms, right? If $(M,\mu,\eta)$ is a monoid, then is $(M\otimes M,\mu\otimes\mu,\eta\otimes\eta)$ also a monoid?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 2nd 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThe multiplication on $M \otimes M$ is supposed to be $$M \otimes M \otimes M \otimes M \stackrel{1 \otimes \sigma \otimes 1}{\to} M \otimes M \otimes M \otimes M \stackrel{\mu \otimes \mu}{\to} M \otimes M,$$ right? The unit can be taken as $\eta \otimes \eta$. The claim in the Hopf monoid article looks wrong to me.

   The multiplication on $M \otimes M$ is supposed to be

   $$M \otimes M \otimes M \otimes M \stackrel{1 \otimes \sigma \otimes 1}{\to} M \otimes M \otimes M \otimes M \stackrel{\mu \otimes \mu}{\to} M \otimes M,$$

   right? The unit can be taken as $\eta \otimes \eta$.

   The claim in the Hopf monoid article looks wrong to me.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeJun 2nd 2014
   • PermaLink
   Author: zskoda
   Format: MarkdownItexYes, something is wrong. Mike and Urs edited the page [[Hopf monoid]] historically, maybe they had some unfinished (unexpressed) idea in mind which would complete to a correct statement. Mike ? Urs ?

   Yes, something is wrong. Mike and Urs edited the page Hopf monoid historically, maybe they had some unfinished (unexpressed) idea in mind which would complete to a correct statement. Mike ? Urs ?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 2nd 2014
   • (edited Jun 2nd 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAll I added to the entry is the sentence on Tannaka duality, some subsection headlines and some pointers to related entries. And I don't have time to look into anything else. But for you guys it will be a breeze to just fix it.

   All I added to the entry is the sentence on Tannaka duality, some subsection headlines and some pointers to related entries. And I don’t have time to look into anything else. But for you guys it will be a breeze to just fix it.

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeJun 2nd 2014
   • (edited Jun 2nd 2014)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI do not know the subject of bimonoids in non-symmetric (and non-braided) monoidal categories. I heard of some work and I do understand the appearance of distributive laws and roughly what fusion is, but it is not my subject. Sorry.

   I do not know the subject of bimonoids in non-symmetric (and non-braided) monoidal categories. I heard of some work and I do understand the appearance of distributive laws and roughly what fusion is, but it is not my subject. Sorry.

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 2nd 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexOf course one can deal with bimonoid structures in a monoidal category, *provided* that those structures really live in the center of the monoidal category. I'll go in and fix the statement if no one says anything more.

   Of course one can deal with bimonoid structures in a monoidal category, provided that those structures really live in the center of the monoidal category. I’ll go in and fix the statement if no one says anything more.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJun 2nd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, Todd!!

   Thanks, Todd!!

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeJun 2nd 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexTodd #2 is right. And yes, I agree, a plain monoidal category is not enough; my bad I guess.

   Todd #2 is right. And yes, I agree, a plain monoidal category is not enough; my bad I guess.

9. • CommentRowNumber9.
   • CommentAuthorzskoda
   • CommentTimeJun 2nd 2014
   • PermaLink
   Author: zskoda
   Format: MarkdownItexTodd -- are you saying that fusions are precisely to make it so ?

   Todd – are you saying that fusions are precisely to make it so ?

10. • CommentRowNumber10.
    • CommentAuthorjoe.hannon
    • CommentTimeJun 3rd 2014
    • PermaLink
    Author: joe.hannon
    Format: MarkdownItexTodd, what is $\sigma$? The antipode? That's fine for a Hopf monoid, but for a bimonoid?

    Todd, what is $\sigma$? The antipode? That’s fine for a Hopf monoid, but for a bimonoid?

11. • CommentRowNumber11.
    • CommentAuthorjoe.hannon
    • CommentTimeJun 3rd 2014
    • PermaLink
    Author: joe.hannon
    Format: MarkdownItexOh, $\sigma$ is the switcheroo. Right.

    Oh, $\sigma$ is the switcheroo. Right.

12. • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 4th 2014
    • (edited Jun 4th 2014)
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexZoran (#9), I don't speak the language of fusion, so I don't know how to respond. But looking at [[Drinfeld center]], it seems to me that could be right.

    Zoran (#9), I don’t speak the language of fusion, so I don’t know how to respond. But looking at Drinfeld center, it seems to me that could be right.