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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?
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?
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?
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.
Re #3: yes.
Re #4: The first one doesn’t embed $Cat$ but rather $Adj$.
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.
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