Timeline for How are the unit/counit of a Hopf algebra and of an categorical adjunction related?
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8 events
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| Oct 11, 2018 at 1:44 | comment | added | Noah Snyder | Right, there is such a thing as a Hopf monad, but they don't have anything to do with adjoint functors. Biadjoint functors give a Frobenius monad but not a Hopf monad. | |
| Oct 10, 2018 at 23:01 | comment | added | AHusain | Plus mentioning that you can make a Hopf monad too. | |
| Oct 10, 2018 at 21:24 | comment | added | Sam Gunningham | There is a general notion of algebra or coalgebra object in a monoidal categeory. Algebras have units (at least the unital ones do) and coalgebras have counits. A Hopf algebra is both an algebra and a coalgebra object in vector spaces. On the other hand, an adunction gives rise to a monad (and a comonad) which are (co)algebra objects in a category of endofunctors - the (co)unit mentioned above is part of this structure. So in this sense they are both examples of a more general notion. I'm not sure if there is anything much deeper than that though. | |
| Oct 10, 2018 at 20:17 | comment | added | Mike Pierce | @QiaochuYuan Gut feeling? Maybe Noah is right and there isn't much of a relationship here. | |
| Oct 10, 2018 at 20:04 | comment | added | Noah Snyder | I don't think they're closely related. The unit/counit of a Frobenius algebra, on the other hand, is somewhat related to adjunctions because LR is a Frobenius monad when L and R are biadjoint. | |
| Oct 10, 2018 at 20:00 | comment | added | Qiaochu Yuan | How do you know that they must be the same idea? | |
| Oct 10, 2018 at 18:12 | history | edited | Mike Pierce | CC BY-SA 4.0 |
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| Oct 10, 2018 at 18:05 | history | asked | Mike Pierce | CC BY-SA 4.0 |