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.
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.
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.
