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Let $(\mathcal A, +,\otimes,I)$ a small symmetric monoidal abelian category. I know that $\mathcal A$ can be embedded into the category of $R$-module for a certain ring $R$. But can we make such embedding also preserve the monoidal structure? More precisely:

Does there exist a commutative ring $R$ such that we can find a strong monoidal fuctor from $\mathcal A$ to $_R\mathsf{Mod}$ which is faithful, full and exact?

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According to this MO question, and more precisely this answer by Jacob Lurie, the answer is in general no.

To sum up the answer, if you have a monoidal functor $F : \mathcal{A} \to \mathsf{Mod}_R$, then $F(I) \cong R$, and thus when $F$ if fully faithful you can recover $R$ as $\operatorname{End}(I)$. This then implies that $F$ is given by $F(A) = \operatorname{Hom}(I,A)$. But there are example of abelian categories for which this isn't a fully faithful embedding, for example the category of complex representations of a finite group.

Theo Johnson-Freyd however mentions a related result: the abelian category can be fully faithfully embedded in the category of bimodules over $R$, by a result of Phùng Hô Hài.

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  • $\begingroup$ Nice example, thanks. $\endgroup$ – Censi LI Jun 30 '16 at 13:57

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