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So in my book, strong monoidal implies preservation of the monoidal unit up to coherent isomorphism. In that case, wouldn’t the forgetful functor $Vect_\mathbb{Q} \to Ab$ be an example?
However, if we drop the condition of preservation of the monoidal unit and just ask that the structural constraint $F A \otimes F B \to F(A \otimes B)$ be an isomorphism, then I’m also struggling.
Yes, to be clear: I am looking for functors between closed monoidal categories that
If we drop the first constraint, it is also not hard to find examples (for example, multiplication by an element $g$ in a group $G$, seen as a monoidal category with only equalities as morphisms).
Oh, I see: preservation of the unit is also implied by being strong closed. Got it.
I think I’ve found an example, although it is a bit contrived. Maybe it reminds you of something more natural.
Take $C$ to be $\mathbb{Z}_2$ as a strict symmetric monoidal discrete category, that is, $C$ has only two objects $0$ and $1$, only identity morphisms, and $0 + 0 = 1 + 1 = 0$, $0 + 1 = 1$.
Take $D$ to be the strict symmetric monoidal poset with natural numbers as objects, sum as monoidal product, and a single morphism $k + 2n \to k$ for all $k, n \geq 0$. This is closed (I think) with $[k,j] = j-k$ if $j \geq k$, $0$ if $k = j+2n$, and $1$ if $k = j+2n+1$. You can also see this as the “strictly commutative” PROP generated by a morphism $2 \to 0$.
Then the inclusion of $0$ and $1$ as objects of $C$ into $D$ is lax monoidal, with the morphism $2 \to 0$ as the only non-identity structural morphism, but it is strictly closed.
I think a general class of examples should come from the inclusions of reflective exponential ideals. Day’s reflection theorem implies that if $C$ is a closed monoidal category and $E\subseteq C$ is a reflective subcategory that is an exponential ideal in the monoidal sense (i.e. $x\in C$ and $y\in E$ imply $[x,y]\in E$), then $E$ is a closed monoidal category with the induced internal-hom and a reflected tensor product, so that its inclusion functor preserves internal-homs but is only lax monoidal (in contrast to its left adjoint, the reflection, which is strong monoidal). It won’t in general preserve the unit strongly, but I think there should be plenty of cases when it does.
For instance, let $C = P A$ be a presheaf category with a Day convolution monoidal structure induced by a monoidal structure on a small category $A$, and let $E$ be the subcategory of $\Phi$-continuous presheaves for some set $\Phi$ of colimits in $A$ that are preserved on both sides by the tensor product (e.g. if $A$ is itself closed). Then $E$ is a reflective exponential ideal, and contains the unit object since the latter is a representable presheaf (at the unit object of $A$) hence preserves all colimits.
Thanks Mike, that’s a great answer!
I added it to the list of examples at closed functor.
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