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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2013

    added to (infinity,n)-category of spans a pointer to the discussion of (,2)-categories of spans in section 10 of

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2013

    added in the Definition section a brief mentioning of the observation in

    that

    Spann(H)EnAlgb(Hop).
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2013

    added a brief comment on how to see the above in an elementary way at least for just 1-fold correspondences.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 7th 2013
    • (edited Aug 8th 2013)

    I need to think more about the following, but this thought is haunting me, so I thought I’d see if anyone can help me with a comment.

    For any category (-category) 𝒞, write 𝒞n for the category of n-dimensional cube diagrams in 𝒞. This is naturally an n-category ((,n)-category) under pasting of cubes.

    Moreover for 𝒞 a category (-category) with pullbacks, write Corr(𝒞) for the category (-category) of objects and correspondences (spans) in 𝒞.

    Now consider the n-category ((,n)-category)

    Corr(𝒞)n

    of n-fold cubes with edges made of correspondences in 𝒞.

    The thought that is haunting me is:

    Isn’t that much like what we would want anything denoted “Corrn(𝒞)” to be?

    Heuristically, an n-cell in Corrn(𝒞) is a bunch of objects, with correspondences between them, correspondences of correspondences between those, and so on, up to n-fold correspondences.

    But that vague description certainly also applies to Corr(𝒞)n.

    You could help me if you can set me straight and point out a good reason why considering Corr(𝒞)n as “the n-category of n-fold correspondences in 𝒞” is a bad idea . If it indeed is.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeAug 8th 2013

    Can you explain more about what is “the category of n-dimensional cube diagrams in 𝒞”?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 8th 2013
    • (edited Aug 8th 2013)

    I just mean: the functor category out of the abstract cellular n-cube n regarded as a poset, regarded as a category. So for n=1 the arrow category, for n=2 the category of commutative squares, etc.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2014

    added statement of the phased tensor product and expanded the discussion of full dualizability accordingly.