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1. • CommentRowNumber1.
   • CommentAuthorKevin Yin
   • CommentTimeAug 30th 2017
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   Author: Kevin Yin
   Format: MarkdownItexHello all, I was examining the adjunction between the tensor and internal hom functors of closed monoidal categories when I realized that a closed category, with the addition of * A transformation $d^X_Y \colon X \to [[X,Y],Y]$ natural in $X$ and extranatural in $Y$ such that * The following diagram commutes for all $X,Y,Z$ (first hexagon identity analogue) $$ \array { X & \stackrel{d^Z_X}{\to} & [[X,Z],Z] & \stackrel{L^{[X,Y]}_{[X,Z] Z}}{\to} & [[[X,Y],[X,Z]],[[X,Y],Z]] \\ \downarrow^{d^{[[X,Y],Z]}_X} &&&& \downarrow^{[L^X_{Y Z},1]} \\ [[X,[[X,Y],Z]],[[X,Y],Z]]] & \stackrel{[[d^Y_X,1],1]}{\to} & [[[[X,Y],Y],[[X,Y],Z]],[[X,Y],Z]]] & \stackrel{[L^{[X,Y]}_{Y Z},1]}{\to} & [[Y,Z],[[X,Y],Z]] } $$ * The following diagram commutes for all $X,Y$ (symmetry analogue) $$ \array { & [X,Y] & \stackrel{d^Y_{[X,Y]}}{\longrightarrow} & [[[X,Y],Y],Y] & \\ & {}_{1_{[X,Y]}}\searrow && \swarrow_{[d^Y_X,1]} & \\ && [X,Y] && } $$ should be enough for the tensor in the left adjoint of the internal hom (if it exists) to form a symmetric (braided) monoidal category, thus constituting a symmetric closed monoidal category. In addition, the transformation $R^Z_{X Y}$ described [here](https://nforum.ncatlab.org/discussion/4632/closed-category/) can be defined as $R^Z_{X Y} = d^{[X,Y]}_{[X,Z]}; [L^X_{Y Z},1]$. Like how symmetric monoidal categories become closed under the adjunction, this "symmetric" closed category would become monoidal. Due to my inexperience I was unable to derive some things such as the second hexagon identity, so I am wondering if such categories are expounded upon in the literature. I would be very grateful if anyone could link me to a paper, or inform me of the proper name of these categories.

   Hello all,

   I was examining the adjunction between the tensor and internal hom functors of closed monoidal categories when I realized that a closed category, with the addition of

   • A transformation $d^X_Y \colon X \to [[X,Y],Y]$ natural in $X$ and extranatural in $Y$

   such that

   • The following diagram commutes for all $X,Y,Z$ (first hexagon identity analogue)

   $$\array { X & \stackrel{d^Z_X}{\to} & [[X,Z],Z] & \stackrel{L^{[X,Y]}_{[X,Z] Z}}{\to} & [[[X,Y],[X,Z]],[[X,Y],Z]] \\ \downarrow^{d^{[[X,Y],Z]}_X} &&&& \downarrow^{[L^X_{Y Z},1]} \\ [[X,[[X,Y],Z]],[[X,Y],Z]]] & \stackrel{[[d^Y_X,1],1]}{\to} & [[[[X,Y],Y],[[X,Y],Z]],[[X,Y],Z]]] & \stackrel{[L^{[X,Y]}_{Y Z},1]}{\to} & [[Y,Z],[[X,Y],Z]] }$$

   • The following diagram commutes for all $X,Y$ (symmetry analogue)

   $$\array { & [X,Y] & \stackrel{d^Y_{[X,Y]}}{\longrightarrow} & [[[X,Y],Y],Y] & \\ & {}_{1_{[X,Y]}}\searrow && \swarrow_{[d^Y_X,1]} & \\ && [X,Y] && }$$

   should be enough for the tensor in the left adjoint of the internal hom (if it exists) to form a symmetric (braided) monoidal category, thus constituting a symmetric closed monoidal category.

   In addition, the transformation $R^Z_{X Y}$ described here can be defined as $R^Z_{X Y} = d^{[X,Y]}_{[X,Z]}; [L^X_{Y Z},1]$.

   Like how symmetric monoidal categories become closed under the adjunction, this “symmetric” closed category would become monoidal. Due to my inexperience I was unable to derive some things such as the second hexagon identity, so I am wondering if such categories are expounded upon in the literature. I would be very grateful if anyone could link me to a paper, or inform me of the proper name of these categories.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeAug 30th 2017
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   Author: Mike Shulman
   Format: MarkdownItexInteresting, I don't think I've seen quite that before. I would have been more inclined to encode symmetry of a closed category with an equivalence $[X,[Y,Z]] \cong [Y,[X,Z]]$; that way if we identify closed categories with closed multicategories, we would be making the multicategory a symmetric multicategory, so that if it is representable then it would automatically be a symmetric monoidal category.

   Interesting, I don’t think I’ve seen quite that before. I would have been more inclined to encode symmetry of a closed category with an equivalence $[X,[Y,Z]] \cong [Y,[X,Z]]$; that way if we identify closed categories with closed multicategories, we would be making the multicategory a symmetric multicategory, so that if it is representable then it would automatically be a symmetric monoidal category.

3. • CommentRowNumber3.
   • CommentAuthorNoam_Zeilberger
   • CommentTimeAug 30th 2017
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   Author: Noam_Zeilberger
   Format: MarkdownItexHello Kevin, The original notion of "symmetric" closed category of Eilenberg and Kelly instead asked for a family of maps $s_{X,Y,Z} : [X,[Y,Z]] \to [Y,[X,Z]]$ natural in $X,Y,Z$ together with some coherence conditions. I think the alternative definition you propose is very natural, though, and I am also interested where/whether it appears in the literature. There are at least some closely related ideas in this old paper by Day and Laplaza (which also includes the formal definition of symmetric closed categories based on the $s$ maps): * B. J. Day and M. L. Laplaza, On Embedding Closed Categories, _Bull. Austral. Math. Soc._ 18 (1978), 357-371. ([doi](https://doi.org/10.1017/S0004972700008236)) It is about the question of when can you embed a closed category into a monoidal closed category, and they prove that you can do this in the symmetric case (but not quite in the non-symmetric case). For the construction, they use the fact that any symmetric closed category $V$ can be given the structure of a [[promonoidal category]] (and hence its category of presheaves can be given a symmetric monoidal closed structure via convolution), with the profunctor $\otimes : V \times V ⇸ V$ defined by $\otimes(X,Y,Z) = Hom(X,[Y,Z])$. I haven't checked all of the diagrams, but I think that your definition exactly corresponds to asking that this promonoidal structure $\otimes$ be equipped with a symmetry. Of course the definition $\otimes(X,Y,Z) = Hom(X,[Y,Z])$ makes sense also in the non-symmetric case, but the problem (as Day and Laplaza observed) is that associativity only holds up to natural transformation rather than isomorphism, so that the resulting presheaf categories are not actually monoidal. This can alternatively be seen as a "feature", and there's actually been a bunch of recent work (with which I'm only partly familiar) on so-called ["skew"-monoidal categories](https://arxiv.org/abs/1201.4981), which are monoidal categories with relaxed associativity and unitality constraints. Skew-monoidal categories have a very tight connection with so-called "skew-closed" categories, which are a slight generalization of the notion of closed category. This is discussed in a recent paper by Street that you might find interesting: * Ross Street, Skew-closed categories, [arXiv:1205.6522](https://arxiv.org/abs/1205.6522). A natural question is what is the right definition of a "symmetric skew-closed category", and I think the answer is something like what you wrote above. I do not know whether Street (or Steve Lack, who has also been [doing a lot of skew stuff lately](http://maths.mq.edu.au/~slack/papers.html#bottom)) has already considered this question.

   Hello Kevin,

   The original notion of “symmetric” closed category of Eilenberg and Kelly instead asked for a family of maps $s_{X,Y,Z} : [X,[Y,Z]] \to [Y,[X,Z]]$ natural in $X,Y,Z$ together with some coherence conditions. I think the alternative definition you propose is very natural, though, and I am also interested where/whether it appears in the literature. There are at least some closely related ideas in this old paper by Day and Laplaza (which also includes the formal definition of symmetric closed categories based on the $s$ maps):

   • B. J. Day and M. L. Laplaza, On Embedding Closed Categories, Bull. Austral. Math. Soc. 18 (1978), 357-371. (doi)

   It is about the question of when can you embed a closed category into a monoidal closed category, and they prove that you can do this in the symmetric case (but not quite in the non-symmetric case). For the construction, they use the fact that any symmetric closed category $V$ can be given the structure of a promonoidal category (and hence its category of presheaves can be given a symmetric monoidal closed structure via convolution), with the profunctor $\otimes : V \times V ⇸ V$ defined by $\otimes(X,Y,Z) = Hom(X,[Y,Z])$. I haven’t checked all of the diagrams, but I think that your definition exactly corresponds to asking that this promonoidal structure $\otimes$ be equipped with a symmetry.

   Of course the definition $\otimes(X,Y,Z) = Hom(X,[Y,Z])$ makes sense also in the non-symmetric case, but the problem (as Day and Laplaza observed) is that associativity only holds up to natural transformation rather than isomorphism, so that the resulting presheaf categories are not actually monoidal. This can alternatively be seen as a “feature”, and there’s actually been a bunch of recent work (with which I’m only partly familiar) on so-called “skew”-monoidal categories, which are monoidal categories with relaxed associativity and unitality constraints. Skew-monoidal categories have a very tight connection with so-called “skew-closed” categories, which are a slight generalization of the notion of closed category. This is discussed in a recent paper by Street that you might find interesting:

   • Ross Street, Skew-closed categories, arXiv:1205.6522.

   A natural question is what is the right definition of a “symmetric skew-closed category”, and I think the answer is something like what you wrote above. I do not know whether Street (or Steve Lack, who has also been doing a lot of skew stuff lately) has already considered this question.

4. • CommentRowNumber4.
   • CommentAuthorKevin Yin
   • CommentTimeAug 30th 2017
   • PermaLink
   Author: Kevin Yin
   Format: MarkdownItexThank you both for your comments. It seems that $d^X_Y$ is roughly equivalent to $s_{X,Y,Z}$ along with the bijection between $C(X,Y)$ and $C(I,[X,Y])$. I was able to find a few missing axioms by looking at that relation, so the papers you linked were a great help.

   Thank you both for your comments. It seems that $d^X_Y$ is roughly equivalent to $s_{X,Y,Z}$ along with the bijection between $C(X,Y)$ and $C(I,[X,Y])$. I was able to find a few missing axioms by looking at that relation, so the papers you linked were a great help.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeAug 31st 2017
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   Author: Mike Shulman
   Format: MarkdownItexNoam, I haven't looked at that paper, but I must be missing something. Can't we just regard a closed category as a closed multicategory and embed any multicategory in the free monoidal category it generates?

   Noam, I haven’t looked at that paper, but I must be missing something. Can’t we just regard a closed category as a closed multicategory and embed any multicategory in the free monoidal category it generates?

6. • CommentRowNumber6.
   • CommentAuthorNoam_Zeilberger
   • CommentTimeAug 31st 2017
   • PermaLink
   Author: Noam_Zeilberger
   Format: MarkdownItexHmm..yes, that sounds right. (I assume you also implicitly meant to include the Yoneda embedding of the free monoidal category as a final step, to get into a closed monoidal category.) I think I was motivating the Day and Laplaza paper poorly. In the introduction they mention another paper by Laplaza on [embedding closed categories](http://www.jstor.org/pss/1997823) where he gives an alternative embedding that works for any closed category $V$. There, as the target he just took the category $[E,Set]$ of covariant presheaves over the (monoidal) category of endofunctors $E = [V,V]$, with the embedding given by the composition of the (contravariant) functor $A \mapsto [A,-]$ with the (contravariant) Yoneda embedding. That's pretty close to the (more slick) embedding you describe, isn't it, with the category of endofunctors replacing the free monoidal category? (It might be worth mentioning your more slick embedding in the article [[closed+category#EmbedIntoCloseMon|closed categories]], though.) The Day and Laplaza embedding is different because instead of freely adding some monoidal structure, they just use the promonoidal structure which is already implicitly there in the closed category (at least in the symmetric case, and otherwise it is only a "skew" promonoidal structure). I'm not exactly sure what kind of extra "mileage" (if any) this gives you as an embedding -- I was mainly interested in it because of these issues with associativity and symmetry.

   Hmm..yes, that sounds right. (I assume you also implicitly meant to include the Yoneda embedding of the free monoidal category as a final step, to get into a closed monoidal category.)

   I think I was motivating the Day and Laplaza paper poorly. In the introduction they mention another paper by Laplaza on embedding closed categories where he gives an alternative embedding that works for any closed category $V$. There, as the target he just took the category $[E,Set]$ of covariant presheaves over the (monoidal) category of endofunctors $E = [V,V]$, with the embedding given by the composition of the (contravariant) functor $A \mapsto [A,-]$ with the (contravariant) Yoneda embedding. That’s pretty close to the (more slick) embedding you describe, isn’t it, with the category of endofunctors replacing the free monoidal category? (It might be worth mentioning your more slick embedding in the article closed categories, though.)

   The Day and Laplaza embedding is different because instead of freely adding some monoidal structure, they just use the promonoidal structure which is already implicitly there in the closed category (at least in the symmetric case, and otherwise it is only a “skew” promonoidal structure). I’m not exactly sure what kind of extra “mileage” (if any) this gives you as an embedding – I was mainly interested in it because of these issues with associativity and symmetry.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeAug 31st 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexAh yes; I thouht that the free monoidal category on a closed multicategory was already closed, but of course it only has hom-objects whose target comes from the original category rather than being a formal product of such. Fortunately the Yoneda embedding of a monoidal category preserves any existing internal-homs. I don't fully understand your description of the Laplaza paper, and I don't have access to it right now, but it seems to me that any embedding of a closed category in a closed monoidal category that *preserves the closed structure* (which is necessary to call it an "embedding") must somehow "use" that closed structure, whether by way of its underlying promonoidal structure, its underlying multicategory structure, or however else.

   Ah yes; I thouht that the free monoidal category on a closed multicategory was already closed, but of course it only has hom-objects whose target comes from the original category rather than being a formal product of such. Fortunately the Yoneda embedding of a monoidal category preserves any existing internal-homs.

   I don’t fully understand your description of the Laplaza paper, and I don’t have access to it right now, but it seems to me that any embedding of a closed category in a closed monoidal category that preserves the closed structure (which is necessary to call it an “embedding”) must somehow “use” that closed structure, whether by way of its underlying promonoidal structure, its underlying multicategory structure, or however else.