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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeOct 18th 2017

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?

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeOct 31st 2017

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?

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeApr 27th 2020

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?

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeApr 27th 2020

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.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeMay 19th 2020

Re #3: yes.

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

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeMay 20th 2020
• (edited May 20th 2020)

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.