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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 31st 2015
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   Author: Todd_Trimble
   Format: MarkdownItexI imagine that somewhere is recorded the fact that the 2-category of symmetric monoidal categories, strong symmetric monoidal functors, and monoidal natural transformations carries a symmetric monoidal 2-category structure, categorifying the fact that the category of commutative monoids carries a symmetric monoidal category structure. I also imagine that this might be deduced by categorifying the familiar result that the category of algebras for commutative or monoidal monads on $Set$ carries an canonical symmetric monoidal (closed) structure, and applying the categorified result to the monoidal 2-monad (a certain club) on $Cat$ whose algebras are symmetric monoidal categories. Can anyone point me to a good reference for this type of thing?

   I imagine that somewhere is recorded the fact that the 2-category of symmetric monoidal categories, strong symmetric monoidal functors, and monoidal natural transformations carries a symmetric monoidal 2-category structure, categorifying the fact that the category of commutative monoids carries a symmetric monoidal category structure. I also imagine that this might be deduced by categorifying the familiar result that the category of algebras for commutative or monoidal monads on $Set$ carries an canonical symmetric monoidal (closed) structure, and applying the categorified result to the monoidal 2-monad (a certain club) on $Cat$ whose algebras are symmetric monoidal categories. Can anyone point me to a good reference for this type of thing?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 2nd 2015
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   Author: Todd_Trimble
   Format: MarkdownItexI wound up asking around, and got the following reference from Mark Weber, a paper by Hyland and Power: [Pseudo-commutative monads and pseudo-closed 2-categories](https://www.dpmms.cam.ac.uk/~martin/Research/Publications/2002/hp02.pdf). Just to save anyone else who saw my query any trouble.

   I wound up asking around, and got the following reference from Mark Weber, a paper by Hyland and Power: Pseudo-commutative monads and pseudo-closed 2-categories.

   Just to save anyone else who saw my query any trouble.