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The page of ncatlab on group object states that:

A group object in $\mathrm{CRing}^{\mathrm{op}}$ is a commutative Hopf algebra.

Question: Is a (noncommutative) Hopf algebra a group object of some category?
[let assume finite dimensional, if necessary.]

The page of Wikipedia on group object states that:

Hopf algebras can be seen as a generalization of group objects to monoidal categories.

It is not clear to me how this sentence answers the above question.

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Not with their definition, where they assume the underlying category to be cartesian. You can define a notion of "Hopf object" in arbitrary symmetric monoidal categories, where you also need to specify the existence of a coproduct map. It's a matter of taste, but I think this general definition should actually be called "group object", so that Hopf algebras would be group objects in the category of vector spaces. The basic observation is that in a cartesian category, every object have a unique coalgebra structure given by the "diagonal" map so this part of the structure is forced in that case, so that those definitions are compatible.

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Cocommutative Hopf algebras are group objects in the cartesian category of cocommutative coalgebras. There is no such description in the non-cocommutative case. Also since the antipode does not need to be invertible, which is certainly true for group objects.

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    $\begingroup$ Cocommutative, counital coalgebras, though I guess that's a standard assumption. $\endgroup$
    – David Roberts
    May 26, 2021 at 22:56
  • $\begingroup$ @DavidRoberts Yes it's standard. $\endgroup$ May 27, 2021 at 5:13

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