TITLE: Is a Hopf algebra a group object of some category?
QUESTION [8 upvotes]: 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.

REPLY [7 votes]: 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.