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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeOct 18th 2017

    Someone must have already studied the Chu construction Chu(Cat,Set)Chu(Cat,Set) on the cartesian closed monoidal category CatCat with dualizing object SetCatSet\in Cat. But “Chu(Cat,Set)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)Chu(Cat,Set) are a pair of categories and a functor from their product to SetSet. 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 CatCat in Chu(Cat,Set)Chu(Cat, Set):

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

    and

    C(C,Set C,Ev)? 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 CatCat but rather AdjAdj.

    • 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)Chu(Set, 2), hence the questions here.