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Created Hopf monoid, as a home for the fact that the category of modules over such inherits a closed monoidal structure.
As for bimonoid I have added a pointer to Tannaka duality for monoids/algebras.
But, hm, isn’t it more precisely rigid monoidal categories with fiber functor that correspond to Hopf monoids?
Perhaps only if the original category in which module structures live is rigid monoidal (e.g., finite-dimensional vector spaces).
Okay, I see. Do we know of a characterization theorem in general then?
So I think the statement in finite-dim vector space is that rigid tensor categories with fiber functor are precisely, up to equivalence, the categories of modules of Hopf algebras.
How much of that characterization statement can be lifted to more general base monoidal categories?
Guys, there is already a stub Hopf monad as well which I created some time ago.
Urs: I remember that Tannakian reconstruction already when the ground field is replaced by a commutative associative unital ring is very problematic and it is known only in few cases like the reconstruction theorem of Nori (which uses Noetherianess assumption). Bruguières has studied this and some noncommutative generalizations.
Also, there is a generalization of Tannaka theory to bialgebroids/Hopf algebroids which is quite tricky and can be interpreted using monoidal sites:
Mike: I disagree with attitude “the generalization” in the idea section! Namely it is generally accepted in Hopf algebra community that in an arbitrary monoidal category generalizing Hopf algebras via the commutative diagrams is not quite sensitive and deeper internal properties should be considered. I mean there is no question about the case of bimonoid. But for Hopf it is essential to have (or even to require by definition) good properties of the categories of Hopf modules, e.g. ${}^H_H\mathcal{M}^H\cong {}^H\mathcal{M}$, or stronger, the fundamental theorem of Hopf modules, ${}_H\mathcal{M}^H\cong \mathcal{M}$. See
Various generalizations (hence not the generalization) have been studied by a number of people.
there is already a stub Hopf monad as well which I created some time ago.
There is also Hopf monoidal category, for what it’s worth. I have tried to interlink all these Hopfy entries.
(By the way, the entry on Hopf monads doesn’t try to say what a Hopf monad is, currently…)
I disagree with attitude “the generalization” in the idea section! Namely it is generally accepted in Hopf algebra community that in an arbitrary monoidal category generalizing Hopf algebras via the commutative diagrams is not quite [ sensible ] and deeper internal properties should be considered.
I think so, too. I think if one looks at all the literature on Hopfy things and their variants, the overarching organizational principle is Tannaka duality. All those conditions on algebras, bialgebra, trialgebra, etc. are justified by the fact that these are characteristic as inducing certain natural structure on their categories of modules.
(I think the (non-)definition of Hopfish algebra is a good point in case, for instance.)
So I’d be hesitant to speak of “Hopf monoids” if their categories of modules are not rigid with fiber functor. For that is what conceptually really defines Hopf algebraic structure, I think.
By the way, the entry on Hopf monads doesn’t try to say what a Hopf monad is, currently
The entry is supposed to collect all approaches on compatible comonad and monad with some sort of duality/rigidity/antipode. So, in future, each section should have its own definition. The article by Mesablishvili and Wisbauer in 5 gives quite much of a general picture as it knowledgeable refers to earlier works, including of Lack and Bruguières. The fundamental theorem of Hopf modules is a key principle. The Tannaka may be related (the paper by Szlachányi in 5 is explaining analogy between comparison functor for monadicity and Tannaka for structures in the categories of bimodules; fundamental theorem may be hi-brow viewed as a kind of one-point descent along torsors, which is sort of comonadicity for more points (like Schneider’s theorem)).
an arch too far?
is there another guiding principle?
The definition of things like “quasitriangular Hopf algebras” is fairly intricate. But the point is: whatever this structure is, it is precisely what makes the category of modules be rigid braided monoidal, which is plain and simple.
And I’d dare say this is also what counts in applications. Like in all applications of quantum groups to quantum field theories, what counts is that their categories of modules have certain natural properties.
But of course one can define what one wants. I just think if in a situation of intricate algebraic conditions one throws out the only guiding principle there is, then it is very hard to tell where the journey is going.
If you want it to say “a generalization” rather than “the generalization”, that’s fine with me. It’s certainly the most obvious generalization, to a category theorist, and it’s a good and useful one.
is there another guiding principle?
Yes: fundamental theorem on Hopf modules (as I stated above with some of the references). It is a pity Tomasz Maszczyk did not publish on the issue, as he has a very deep understanding of that issue.
If you want it to say “a generalization” rather than “the generalization”
Exactly.
is there another guiding principle?
Yes: fundamental theorem on Hopf modules
Okay, so for which ambient monoidal categories is this true for Hopf monoids, and for which definition of Hopf monoids?
That is itself one possible definition: the bimonoid is a Hopf monoid iff the fundamental theorem of Hopf modules holds. True for example for the ambient category of modules over a commutative associative unital ring.
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