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    • CommentRowNumber101.
    • CommentAuthoratmacen
    • CommentTimeNov 7th 2018

    Re #99: Thanks.

    Re #100:

    I’m not arguing for (i) over (ii), but I don’t see how (ii) in [C op,Set][C^op,Set] is a way of talking about (i) in CC. I don’t know what (i) in CC even means, since CC may not have a universe. There seems to be a serious misunderstanding about what (i) and (ii) are.

    • CommentRowNumber102.
    • CommentAuthorMike Shulman
    • CommentTimeNov 7th 2018

    The partiality of (i) in CC is also external, so it doesn’t need a universe.

    • CommentRowNumber103.
    • CommentAuthoratmacen
    • CommentTimeNov 7th 2018

    I reread some earlier comments on this subthread about (i) vs (ii). I think you’re right, and I misunderstood what Dan wants. It now seems to me like he might count natural families ΠX:Ob(C),(Hom(yX,Tm)) \Pi X\,:\,Ob(C),\,(Hom(y X,Tm))_\bot as (i). And as far as I understand, this is essentially what you get from an element of Tm Tm_\bot constructed in the semantic logical framework without funny tricks.