(Alternate title: Find the Adjoint: Hopf Algebra edition)
I was chatting with Jonah about his question Hopf algebra structure on $\prod_n A^{\otimes n}$ for an algebra $A$. It's very closely related to the following question:
For a $k$-algebra $A$, is there a Hopf Algebra $H(A)$ such that for any Hopf algebra $B$, we have
$\mathop{Hom}_{k\text{-alg}}(B,A)\cong \mathop{Hom}_{k\text{-Hopf-alg}}(B,H(A))$?
In other words, does the forgetful functor from $k$-Hopf-algebras to $k$-algebras have a right adjoint? There are a few related questions, any of which I'd be interested in knowing the answer to:
- Does the forgetful functor from Hopf algebras to augmented algebras (sending the counit to the augmentation) have a right adjoint?
- Does the forgetful functor from Hopf algebras to algebras with distinguished anti-automorphism (sending the antipode to the anti-automorphism) have a right adjoint?
- Does the forgetful functor from Hopf algebras to algebras with augmentation and distinguished anti-automorphism have a right adjoint?
Unfortunately, I don't feel like I can motivate this question very well. My motivation is that the better I know which forgetful functors have adjoints, the better I sleep at night.
Best Answer
The article here proves that the forgetful functor from $k$-Hopf algebras to $k$-algebras has a right adjoint. The main tool they use is the Special Adjoint Functor Theorem.