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    • CommentRowNumber1.
    • CommentAuthorGuest
    • CommentTimeOct 14th 2009
    Comment at (n,r)-category about the equivalence of fundamental categories. This is a coarser notion than equivalence of categories in the usual sense.

    -David Roberts
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 14th 2009

    Thanks. Do you have a reference for that notion? Is that in Grandis' work? We should have an entry on this.

    Recently, after I carried this question to the CatTheory mailing list I received a reply by Peter Bubenik who wrote that together with David Spivak they are in the process of proving the "directed homotopy hypothesis" relating (oo,1)-categories and some flavor of directed topological spaces.

    But even though I tried, I couldn't make him tell me what exactly it is they are proving and precisely which notions of equivalence etc they are using.

    • CommentRowNumber3.
    • CommentAuthorEric
    • CommentTimeNov 16th 2009

    Asked a question at (n,r)-category.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 16th 2009

    I have replied.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 3rd 2011
    • (edited Jun 3rd 2011)

    I have added to the Definition-section at (n,r)-category a precise definition:

    In terms of the standard notion of (∞,n)-categories we can make this precise as follows:

    For 2n-2 \leq n \leq \infty, an (n,0)-category is an ∞-groupoid that is n-truncated: an n-groupoid.

    For 0r<0 \leq r \lt \infty, an (n,r)-category is an (∞,r)-category CC such that for all objects X,YCX,Y \in C the (,r1)(\infty,r-1)-categorical hom-object C(X,Y)C(X,Y) is an (n1,r1)(n-1,r-1)-category.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 1st 2018

    Added a reference.

    diff, v49, current

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 1st 2018

    Is there anything in those query boxes worth keeping?

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeSep 1st 2018

    I think their conclusions should be incorporated into the page.

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