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1. • CommentRowNumber1.
   • CommentAuthoramarh
   • CommentTimeFeb 11th 2018
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   Author: amarh
   Format: TextLax monoidal and (lax) closed functors between closed monoidal categories are essentially the same thing, and there is plenty of examples of strong monoidal functors that are only lax closed. However, I have been struggling to think of an example of a strong closed functor between closed monoidal categories that is not strong monoidal. Has anyone here encountered one of these "in nature"? I can see why the first situation would be much more common with cartesian closed categories, as "preserving finite products" happens under simple conditions, but I don't see why one should be more common than the other with general monoidal closed categories.

   Lax monoidal and (lax) closed functors between closed monoidal categories are essentially the same thing, and there is plenty of examples of strong monoidal functors that are only lax closed. However, I have been struggling to think of an example of a strong closed functor between closed monoidal categories that is not strong monoidal.
   
   Has anyone here encountered one of these "in nature"?
   
   I can see why the first situation would be much more common with cartesian closed categories, as "preserving finite products" happens under simple conditions, but I don't see why one should be more common than the other with general monoidal closed categories.
2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 11th 2018
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   Author: Todd_Trimble
   Format: MarkdownItexSo 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthoramarh
   • CommentTimeFeb 11th 2018
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   Author: amarh
   Format: MarkdownItexYes, to be clear: I am looking for functors between closed monoidal categories that * preserve the unit up to coherent isomorphism, * preserve internal homs up to coherent isomorphism, * do not preserve monoidal products. 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).

   Yes, to be clear: I am looking for functors between closed monoidal categories that

   • preserve the unit up to coherent isomorphism,
   • preserve internal homs up to coherent isomorphism,
   • do not preserve monoidal products.

   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).

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 11th 2018
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   Author: Todd_Trimble
   Format: MarkdownItexOh, I see: preservation of the unit is also implied by being strong closed. Got it.

   Oh, I see: preservation of the unit is also implied by being strong closed. Got it.

5. • CommentRowNumber5.
   • CommentAuthoramarh
   • CommentTimeFeb 12th 2018
   • (edited Feb 12th 2018)
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   Author: amarh
   Format: MarkdownItexI 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 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.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 12th 2018
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   Author: Mike Shulman
   Format: MarkdownItexI 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.

   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.

7. • CommentRowNumber7.
   • CommentAuthoramarh
   • CommentTimeFeb 12th 2018
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   Author: amarh
   Format: MarkdownItexThanks Mike, that's a great answer!

   Thanks Mike, that’s a great answer!

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 12th 2018
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   Author: Mike Shulman
   Format: MarkdownItexI added it to the list of examples at [[closed functor]].

   I added it to the list of examples at closed functor.