# Monoidal adjunction whose right-adjoint functor has structure morphisms which are epimorphisms

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.

• There is an obvious typo: $R(X \otimes R)$ should instead be $R(X \otimes Y)$. – Geoffrey Trang May 26 '20 at 23:00
• Thanks. I'm looking now and I think that @jgon have fixed it. – Math-Phys-Cat Group May 27 '20 at 8:09

## 1 Answer

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.

• 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. – Math-Phys-Cat Group May 27 '20 at 8:11