The page of ncatlab on [group object](https://ncatlab.org/nlab/show/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](https://en.wikipedia.org/wiki/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.



Is a Hopf algebra a group object of some category?
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.


