Adjoining an antipode to a bialgebra

Tannaka duality allows the possibility of taking an R-bialgebra A, applying some categorical construction to $\Comod_R(A)\cauchy$, and asking whether the result again has the form $\Comod_R(B)\cauchy$ for some R-bialgebra B.

An example of an appropriate categorical construction is adjoining left-dual objects to a tensor category. To each tensor category $\,\scrC$, there is a left-autonomous tensor category $\auton_\ell(\scrC)$ and a tensor functor $\functor\from \,\scrC \to {\auton_\ell(\scrC)}$ which induces a natural equivalence between the category of tensor functors $\functor\from \auton_\ell(\scrC) \to \scrD$ and the category of tensor functors $\functor\from \,\scrC \to \scrD$ for all left autonomous tensor categories $\scrD$. (See [JS91a] and paper II in the series.)

Suppose that $\functor F \from \,\scrC \to {\Mod_R}$ is a tensor functor whose values $\FX$ are cauchy R-modules. Then we obtain a corresponding tensor functor $\functor \hat F \from \,\auton_\ell(\scrC) \to {\Mod_R}$.

Proposition 18.1   $E_{\!\hat F}^\pprime\,$ is the reflection of the R-bialgebra $E_{\!F}^\pprime\,$ into the category of Hopf R-algebras.

This gives a construction for adjoining an antipode to a bialgebra over a field R; that is, a construction for a left adjoint to the inclusion of the category $\Hopf_R$ of Hopf algebras in the category $\BIG_R$ of bialgebras. Given a bialgebra A, put $F=\map U_{\!A} \from \Comod_R(A)\cauchy\to {\Mod_R}$. By the proposition from the previous section, we have $A\isom E_{\!F}^\pprime$. By the above result, the Hopf algebra $H=E_{\!\hat F}^\pprime$ is the required reflection.

If we require the adjoined antipode to be invertible, we must replace $\auton_\ell(\scrC)$ in the above by $\auton\!(\scrC)$ which is the free autonomous tensor category on the tensor category $\,\scrC$. And so on.