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1. • CommentRowNumber1.
   • CommentAuthorFinnLawler
   • CommentTimeMay 13th 2011
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   Author: FinnLawler
   Format: MarkdownItexJust a quick question, in case anyone can tell me off hand (before I go scouring the literature): does the Gray tensor product make the category of strict 2-categories and pseudofunctors into a monoidal closed category? If so, is there a name for categories enriched in this category?

   Just a quick question, in case anyone can tell me off hand (before I go scouring the literature): does the Gray tensor product make the category of strict 2-categories and pseudofunctors into a monoidal closed category? If so, is there a name for categories enriched in this category?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMay 14th 2011
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   Author: Mike Shulman
   Format: MarkdownItexI haven't seen this written down, I don't think, or checked the details, but I'd be surprised if it weren't monoidal and I'd be surprised if it *were* closed. For instance, since 1 is the unit for the Gray tensor product, you'd have a bijection between pseudofunctors $A\to B$ and pseudofunctors $1\to Hom(A,B)$. But the latter is not just an object of $Hom(A,B)$ but an automorphism in it coherently isomorphic to the identity. So I don't see a natural candidate for $Hom(A,B)$.

   I haven’t seen this written down, I don’t think, or checked the details, but I’d be surprised if it weren’t monoidal and I’d be surprised if it were closed. For instance, since 1 is the unit for the Gray tensor product, you’d have a bijection between pseudofunctors $A\to B$ and pseudofunctors $1\to Hom(A,B)$. But the latter is not just an object of $Hom(A,B)$ but an automorphism in it coherently isomorphic to the identity. So I don’t see a natural candidate for $Hom(A,B)$.

3. • CommentRowNumber3.
   • CommentAuthorFinnLawler
   • CommentTimeMay 19th 2011
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   Author: FinnLawler
   Format: MarkdownItexThanks for that, Mike. I thought I might need this to define some sort of 3-category of biprofunctors, but I don't think it works. (It seems Biprof is best defined just as a subtricategory of Bicat.)

   Thanks for that, Mike. I thought I might need this to define some sort of 3-category of biprofunctors, but I don’t think it works. (It seems Biprof is best defined just as a subtricategory of Bicat.)