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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeOct 18th 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexSomeone must have already studied the [[Chu construction]] $Chu(Cat,Set)$ on the cartesian closed monoidal category $Cat$ with dualizing object $Set\in Cat$. But "$Chu(Cat,Set)$" is kind of hard to seach for, and right now I can't find anything about it. Does anyone know a reference?

   Someone must have already studied the Chu construction $Chu(Cat,Set)$ on the cartesian closed monoidal category $Cat$ with dualizing object $Set\in Cat$. But “$Chu(Cat,Set)$” is kind of hard to seach for, and right now I can’t find anything about it. Does anyone know a reference?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeOct 31st 2017
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   Author: Mike Shulman
   Format: MarkdownItexAh, finally found something: [a cat-list discussion from 2006](http://www.mta.ca/~cat-dist/archive/2006/06-5). Of course, it has to be a 2-categorical Chu construction with morphisms that are adjoint up to isomorphism rather than equality. But it doesn't seem like anyone took it anywhere after that brief exchange?

   Ah, finally found something: a cat-list discussion from 2006. Of course, it has to be a 2-categorical Chu construction with morphisms that are adjoint up to isomorphism rather than equality. But it doesn’t seem like anyone took it anywhere after that brief exchange?

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 27th 2020
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSo the elements of $Chu(Cat,Set)$ are a pair of categories and a functor from their product to $Set$. Up to some 'op' issues, aren't these just profunctors?

   So the elements of $Chu(Cat,Set)$ are a pair of categories and a functor from their product to $Set$. Up to some ’op’ issues, aren’t these just profunctors?

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 27th 2020
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWhat's to be said about two ways to embed $Cat$ in $Chu(Cat, Set)$: $$ C \mapsto (C,C^{op},Hom) $$ and $$ C \mapsto (C, Set^C, Ev) ? $$ I guess there's yoneda on the second component.

   What’s to be said about two ways to embed $Cat$ in $Chu(Cat, Set)$:

   $$C \mapsto (C,C^{op},Hom)$$

   and

   $$C \mapsto (C, Set^C, Ev) ?$$

   I guess there’s yoneda on the second component.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeMay 19th 2020
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexRe #3: yes. Re #4: The first one doesn't embed $Cat$ but rather $Adj$.

   Re #3: yes.

   Re #4: The first one doesn’t embed $Cat$ but rather $Adj$.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 20th 2020
   • (edited May 20th 2020)
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   Author: David_Corfield
   Format: MarkdownItexThanks! Gosh, I was just thinking that you were responding to some old questions of mine when I noticed they were asked only 23 days ago. It seem like a lifetime ago I asked, during a brief foray into a possible 2-Isbell duality. Must be the time-bending effects of lockdown. I was wondering back then about the relationship between 1-Isbell duality and $Chu(Set, 2)$, hence the questions here.

   Thanks!

   Gosh, I was just thinking that you were responding to some old questions of mine when I noticed they were asked only 23 days ago. It seem like a lifetime ago I asked, during a brief foray into a possible 2-Isbell duality. Must be the time-bending effects of lockdown.

   I was wondering back then about the relationship between 1-Isbell duality and $Chu(Set, 2)$, hence the questions here.