# Hopf Algebras/Rings, A Question of Terminology

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?

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That's what the antipode is for, right? –  Qiaochu Yuan Dec 2 '10 at 17:04

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.
Robin, no, you need a diagonal to even state the axiom for inverses. But you can define group objects in monoidal categories with a diagonal. That's what Ravenel and Wilson mean. Todd, does it also worry you that if you want to define a ring as a monoid in abelian groups, i.e. as a map $A \otimes A \to A$, that $\otimes$ is not the product in abelian groups? –  Tilman Dec 2 '10 at 18:44