I don't know if this somewhat vague question has a precise answer, but I'd appreciate thoughts and references.
Let $k$ be a field. I am looking at Hopf algebras over $k$ by which I mean an algebra and coalgebra $(H, \mu , \Delta )$ equipped with an antipode $S : H \to H$ satisfying the usual compatibility and antipode axioms.
I'm wondering what the natural notion of "(co)free-ness" is in this setting? Or perhaps what is a good replacement for this question? For instance, should I rather ask for free/cofree bialgebras?
I am interested in (co)freeness relative to any base category, for example "free Hopf algebra generated by a set" or "free Hopf algebra generated by a vector space"...
I guess it is the mixture of algebra and coalgebra that is throwing me off. I don't know what kind of universal property to ask for.
From algebras (and similar algebraic gadgets) I am used to free objects being those that arise as the values of a left adjoint to a forgetful functor. From wikipedia I see that cofree objects arise as the values of a right adjoint to the forgetful functor. So perhaps a free and cofree Hopf algebra is one which has both of the universal properties of free algebras and cofree coalgebras.
One possibility (following Takeuchi) is as follows: Let $C$ be a coalgebra over a field $k$ . The free Hopf algebra $(H(C), i)$ generated by $C$ is characterized by the following universal property: (1) $i:C\rightarrow H(C)$ is a coalgebra map (2) $Hom(i, H)$ : Hopf $(H(C), H)\rightarrow Coalg(C, H)$ is a bijection for any Hopf algebra $H$.