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1. • CommentRowNumber1.
   • CommentAuthorChris SchommerPries
   • CommentTimeMar 11th 2014
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   Author: Chris SchommerPries
   Format: MarkdownItex[globular operads](http://ncatlab.org/nlab/show/globular+operad) are a great way to formalize all kinds of higher categorical structure you can put on globular sets. I am wondering if there is a globular operad corresponding to the theory of symmetric monoidal categories (and similar higher dimensional versions). I am worried because the n-lab page says that the resulting monads on globular sets can be characterized as those which are Cartesian and also over T, the free strict n-category monad. I fear that the symmetric monoidal category monad might fail to be Cartesian for the same reason that the commutative monoid monad on sets fails to be Cartesian. What is to be done about this?

   globular operads are a great way to formalize all kinds of higher categorical structure you can put on globular sets. I am wondering if there is a globular operad corresponding to the theory of symmetric monoidal categories (and similar higher dimensional versions).

   I am worried because the n-lab page says that the resulting monads on globular sets can be characterized as those which are Cartesian and also over T, the free strict n-category monad. I fear that the symmetric monoidal category monad might fail to be Cartesian for the same reason that the commutative monoid monad on sets fails to be Cartesian.

   What is to be done about this?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMar 11th 2014
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   Author: Mike Shulman
   Format: MarkdownItexI think the symmetric monoidal category monad *is* cartesian; it precisely remedies the problem with the commutative monoid monad by causing $a\otimes b$ to be only isomorphic to $b\otimes a$ rather than equal to it.

   I think the symmetric monoidal category monad is cartesian; it precisely remedies the problem with the commutative monoid monad by causing $a\otimes b$ to be only isomorphic to $b\otimes a$ rather than equal to it.

3. • CommentRowNumber3.
   • CommentAuthorChris SchommerPries
   • CommentTimeMar 12th 2014
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   Author: Chris SchommerPries
   Format: MarkdownItexI see Lienster's book mentions something along these lines for symmetric strict monoidal categories. I wish there was an easy way to check this.

   I see Lienster’s book mentions something along these lines for symmetric strict monoidal categories. I wish there was an easy way to check this.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMar 12th 2014
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   Author: Mike Shulman
   Format: MarkdownItexEasier than just checking the definition?

   Easier than just checking the definition?

5. • CommentRowNumber5.
   • CommentAuthorChris SchommerPries
   • CommentTimeMar 12th 2014
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   Author: Chris SchommerPries
   Format: MarkdownItexWait, I don't think there is a map of monads $P \to T$, where T is the strict 2-category monad and P is the symmetric monoidal bicategory monad. Such a map would mean that every strict 2-category could be viewed as a special case of a symmetric monoidal bicategory, which is absurd. So I guess I am asking if there is some alternative theory to globular operads which is similar but includes this example?

   Wait, I don’t think there is a map of monads $P \to T$, where T is the strict 2-category monad and P is the symmetric monoidal bicategory monad. Such a map would mean that every strict 2-category could be viewed as a special case of a symmetric monoidal bicategory, which is absurd.

   So I guess I am asking if there is some alternative theory to globular operads which is similar but includes this example?

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeMar 12th 2014
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   Author: Mike Shulman
   Format: MarkdownItexI don't know.

   I don’t know.