bialgebras and hopf algebras over a category

Question

I am currently looking for a reference on how to define a bialgebra over a category or a Hopf algebra over a category,

I have consulted in several texts of Hopf algebras but only define these structures on rings or fields.

I am grateful for the person who can guide me on this topic.

Answer 1

The difference between Hopf algebra "in", as opposed to "over," something is actually crucial here. A Hopf algebra in a symmetric monoidal category is just an object with monoidal and comonoid structures which are homomorphisms for each other, along with an antipode map satisfying the usual Hopf algebra axioms. The point is that this is an internalization, not a generalization of possible bases to be defined over. So a Hopf algebra in the category of $k$-vector spaces is the same as a Hopf algebra over $k$.