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Let $(\mathbf{C},\otimes,1)$ and $(\mathbf{D},*,e)$ be monoidal categories and let $L:\mathbf{C}\rightarrow \mathbf{D}$ and $R:\mathbf{D}\rightarrow \mathbf{C}$ be functors. Suppose that there exists an adjunction between $L$ and $R$ such that $L$ is the left adjoint and $R$ is the right adjoint. Suppose that $L$ is a strong monoidal functor, so that by the doctrinal adjunction it follows that $R$ is lax monoidal and your structural morphisms $$ \mu_{XY}:R(X)*R(Y)\rightarrow R(X\otimes Y)$$ is determined by the unity $\eta$ and the counity $\epsilon$ of the adjunction between $F$ and $G$ (see here). My question is:

  • under which conditions on $L,R,\mathbf{C}$ and/or $\mathbf{D}$ the structural maps $\mu_{XY}$ are epimorphisms?

P.S: the concrete situation in which I'm insterested is when $\mathbf{C}=\mathbf{Mod}_A$ and $\mathbf{D}=\mathbf{Mod}_B$ are the momonoidal categories of $A$-modules and $B$-modules, where $A,B$ are commutative rings with unit and $A\subset B$ is a subring. Furthermore, $L$ is the extension of scalar functor and $R$ is the restriction of scalar functor.

Any help is welcome. Thanks.

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  • $\begingroup$ There is an obvious typo: $R(X \otimes R)$ should instead be $R(X \otimes Y)$. $\endgroup$ – Geoffrey Trang May 26 '20 at 23:00
  • $\begingroup$ Thanks. I'm looking now and I think that @jgon have fixed it. $\endgroup$ – Math-Phys-Cat Group May 27 '20 at 8:09
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Let's look at your particular case: then you are wondering when $X \otimes_A Y \to X \otimes_B Y$ is a surjection of $A$-modules. The answer here is always! This is just because $X \otimes_B Y$ is the quotient of $X \otimes_A Y$ by the relations $x \otimes b y \sim b x \otimes y \sim b(x \otimes y)$ for all $b \in B$, and the map $X \otimes_A Y \to X \otimes_B Y$ you described is precisely the canonical projection if you trace through the definition. I'll try to think more about the general case, but hopefully this helps for now.

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  • $\begingroup$ Thanks a lot! This will be very helpful in my particular case of interest. But I'm also curious about an answer to the general case. $\endgroup$ – Math-Phys-Cat Group May 27 '20 at 8:11

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