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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 6th 2010

    I got annoyed with the fact that these links did not exist, and so I created now stubs for them:

    To the latter entry I moved the references on (,1)-topos theory that had been lsited at higher topos theory.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeNov 7th 2010

    Mike has in his pages much about his thinking about 2-topoi…

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeNov 7th 2010

    Yeah, and I’ve been planning to move it to the main nLab for a while. But for now, a link will have to do.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 24th 2021
    • (edited Aug 24th 2021)

    Since Joel Hamkins is looking at partial orders at his blog, I was wondering about Pos and its structure. Presumably it should be the archetypal (1,2)-topos, whatever that is.

    Those (non-commutative) ordered sum and product presumably are explicable category-theoretically.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeAug 24th 2021

    The ordered sum is the collage of the terminal profunctor, and also the cocomma object of the product projections AA×BB (a.k.a. the join).

    The best I can do right now for the ordered product is that it’s the (generalized) Grothendieck construction of the lax functor BProf that sends all objects to A and all nonidentity morphisms to the terminal profunctor. Note that well-definedness of the latter depends on antisymmetry of B; it’s not clear to me that it has a generalization to non-posets, or even to other (1,2)-toposes (since the definition as I gave it uses excluded middle).

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 24th 2021

    Ok, thanks! So is there anything to learn from it being a (1,2)-topos, a topic we have absolutely nothing about on nLab.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 25th 2021

    Talking with John Baez elsewhere, seeing these poset composition operations in terms of operads might shed some light.