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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?

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  • $\begingroup$ I don't understand your example in the second paragraph. It's certainly not the case that every algebra is Morita equivalent to a Clifford algebra. I think you're missing some hypotheses? Similarly, it's certainly not the case that every algebra is semisimple. Maybe you could be more explicit about exactly what your definitions are? $\endgroup$ – Qiaochu Yuan Apr 26 '16 at 1:51
  • $\begingroup$ My example didn't make sense, so I've gotten rid of it. I am assuming semisimplicity. $\endgroup$ – Alex Turzillo Apr 26 '16 at 2:16
  • $\begingroup$ Also, depending on what you mean by "semisimple module category," I don't think your claim that these are all categories of modules over algebras is true either, since (with the definitions I have in mind) there may be infinitely many simple objects. $\endgroup$ – Qiaochu Yuan Apr 26 '16 at 3:25
  • $\begingroup$ Right. I mean to refer to the main theorem of arXiv:math/0111139, which relies on additional assumptions. $\endgroup$ – Alex Turzillo Apr 26 '16 at 4:00
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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.

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