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Added to bimonoid the fact that the category of modules over a bimonoid is monoidal.
I added in this section a link to “fiber functor” and then added at the bottome the sentence:
These relations are known as Tannaka duality for monoids/algebras, see at structure on algebras and their module categories - table.
I added a few string diagrams to the entry.
Nice!
is a bimonoid object in Ab a biring?
I don’t think so. There is the explicit axioms for a biring in the appendix A of “Representable functors and operations on rings” (D. 0 . TALL and G. C. WRAITH, 1968).
The only thing true must be that a biring is a monoid object in Ab. It doesn’t seem to be a comonoid object and there is noting like the compatiblity between multiplication and comultiplication which is the most important axiom for a bimonoid.
Maybe unpacking the explicit diagrams for a biring on the page “biring” would be a good idea.
@8 You could probably call them integer bialgebras, as abelian groups are integer modules, and bialgebras are by definition bimonoids in modules.
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