I'm reading $\textit{The Hopf Ring for Complex Cobordism}$ by Ravenel and Wilson where they discuss the notion of group and ring objects over a category. They say that a hopf algebra over a ring in this context is a group object in the category of coalgebras. Here's my problem. I assume that the group operation for a hopf algebra must be the algebra multiplication, since the addition is presupposed from the coalgebra structure. Not all hopf algebras are division algebras. But, shouldn't a group object have inverses?



Hopf Algebras/Rings, A Question of Terminology
The group operation corresponds to the multiplication map $\mu:A\otimes A\to A$
and the identity should be the natural map $\iota:k\to A$. Both these should
be coalgebra maps.
The inverse should correspond to a map $S:A\to A$ with
$\mu\circ(\rm{id}\otimes S)\circ\Delta=\iota\circ\epsilon
=\mu\circ(S\otimes\rm{id})\circ\Delta$, so $S$ is the antipode.


