# Can tensor abelian categories always be embedded into the category of modules?

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?

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.

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