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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2020

    have added pointer to

    diff, v11, current

    • CommentRowNumber2.
    • CommentAuthorHurkyl
    • CommentTimeFeb 22nd 2021
    • (edited Feb 22nd 2021)

    The nerve operation CnCore(C[n]) is a reflective embedding (,1)Cat𝒫(Δ), where 𝒫 means the -category of Gpd-valued presheaves.

    I want to add the following claim here: the nerve factors through the reflective embedding 𝒫(Δ2)𝒫(Δ)?

    The details are subtle so I’m not sure of the result, but I think this is equivalent to every -category being expressible as a colimit of a diagram using only the objects [0],[1],[2], which I believe is true by expressing an -category as a colimit of categories, and rewrite categories as colimits of [0],[1],[2] in a compatible way to make a single diagram. (using just [0] and [1] is not enough)

    Another thing I wanted to add to this entry (is there a better page to talk about this nerve construction?) is a description of (,1)Cat as a subcategory of local objects. If the answer to the above is yes, I believe that implies that the -categories are the objects local with respect to Sp2Δ2, Sp3Δ3, and Δ0J, where Spn is the spine of Δn and J is the indiscrete simplicial space with J0 the set of two elements. (i.e. take the nerve of the contractible groupoid on two elements, and then embed sets in -groupoids)

    I have two questions about trying to simplify that description further:

    • I think I’ve seen things on Rezk completeness that would imply the Sp3Δ3 condition is redundant. Is that true?
    • Can we replace J with the union of two 2-simplices expressing left and right inverses fg1 and gh1? Does this change the answer to the previous question?
    • CommentRowNumber3.
    • CommentAuthorRuneHaugseng
    • CommentTimeFeb 23rd 2021

    What you’re suggesting is not correct: you have to consider the spine inclusions for all n to describe -categories. Otherwise, for instance, any -groupoid would be local for ΔnΔn, which would imply that the (n1)-sphere was contractible. (There is a big difference between “every -category is an iterated colimit of copies of Δ1 and Δ0” (which is true) and “every -category 𝒞 is the colimit of the canonical diagram of shape (Δn)/𝒞” which is what you would need to get a fully faithful embedding in presheaves on Δn - this is false for every finite n.)

    The answer to your final question is “yes”, though - you can definitely also characterize completeness by looking at morphisms with separate left and right inverses.

    • CommentRowNumber4.
    • CommentAuthorHurkyl
    • CommentTimeFeb 23rd 2021
    • (edited Feb 23rd 2021)

    Oh right, it’s not enough that an -category be a colimit of a diagram with objects [0],[1],[2], but it has to be that specific diagram.

    I see now how to bridge the gap I was missing in trying to construct a counterexample. Since an -groupoid G embeds as a constant presheaf, natural transformations KNG correspond to morphisms colimKG, and colim converts a simplicial discrete space to the homotopy type of the corresponding simplicial set.

    • CommentRowNumber5.
    • CommentAuthorHurkyl
    • CommentTimeFeb 24th 2021

    Whoops, forgot to alert I made the edit.

    diff, v13, current