When are Morita classes represented by certain structured algebra objects?

Question

Let $\mathcal{C}$ be a monoidal category. There is a notion of Morita equivalence of algebra objects internal to $\mathcal{C}$. Does each Morita class have a symmetric Frobenius representative? A Hopf representative?

Following arXiv:math/0111139, suppose $\mathcal{C}$ is semisimple and rigid with finitely many irreducible objects and irreducible unit. Each indecomposable semisimple module category over $\mathcal{C}$ is equivalent to one of the form $\text{Mod}_\mathcal{C}A$ for some algebra object $A$ internal to $\mathcal{C}$; such algebras are called indecomposable semisimple, and their Morita classes are in bijective correspondence with such module categories. In this language, my question is whether, given such a module category $\mathcal{M}$ over $\mathcal{C}$, there is a symmetric Frobenius (or Hopf) algebra object $A$ in $\mathcal{C}$ such that $\text{Mod}_\mathcal{C}A$ is equivalent to $\mathcal{M}$.

If not, are there conditions on $\mathcal{C}$ under which $A$ exists?

Answer 1

No. In the category of supervector spaces with the Koszul signs, odd Clifford algebras are neither symmetric Frobenius nor Hopf.

Note that symmetric Frobenius structures transfer across Morita equivalences. Hopf structures do not but you can see that no algebra in the Morita class of an odd Clifford algebra can be Hopf as follows. If A is Hopf, then Mod(A) comes equipped with a C-enriched fiber functor to C, and in particular the monoidal unit of Mod(A) has endomorphism algebra the same as that of C. But in SuperVec, the monoidal unit has endomorphism algebra the ground field whereas Cliff(odd) module has endomorphism algebra containing an odd Clifford algebra.