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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])C \mapsto n \mapsto Core(C^{[n]}) is a reflective embedding (,1)Cat𝒫(Δ)(\infty,1)Cat \to \mathcal{P}(\Delta), where 𝒫\mathcal{P} means the \infty-category of Gpd\infty Gpd-valued presheaves.

    I want to add the following claim here: the nerve factors through the reflective embedding 𝒫(Δ 2)𝒫(Δ)\mathcal{P}(\Delta_{\leq 2}) \to \mathcal{P}(\Delta)?

    The details are subtle so I’m not sure of the result, but I think this is equivalent to every \infty-category being expressible as a colimit of a diagram using only the objects [0],[1],[2][0], [1], [2], which I believe is true by expressing an \infty-category as a colimit of categories, and rewrite categories as colimits of [0],[1],[2][0],[1],[2] in a compatible way to make a single diagram. (using just [0][0] and [1][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(\infty,1)Cat as a subcategory of local objects. If the answer to the above is yes, I believe that implies that the \infty-categories are the objects local with respect to Sp 2Δ 2Sp^2 \to \Delta^2, Sp 3Δ 3Sp^3 \to \partial \Delta^3, and Δ 0J\Delta^0 \to J, where Sp nSp^n is the spine of Δ n\Delta^n and JJ is the indiscrete simplicial space with J 0J_0 the set of two elements. (i.e. take the nerve of the contractible groupoid on two elements, and then embed sets in \infty-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 Sp 3Δ 3Sp^3 \to \partial \Delta^3 condition is redundant. Is that true?
    • Can we replace JJ with the union of two 2-simplices expressing left and right inverses fg1fg \simeq 1 and gh1gh \simeq 1? 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 nn to describe \infty-categories. Otherwise, for instance, any \infty-groupoid would be local for Δ nΔ n\partial \Delta^n \to \Delta^n, which would imply that the (n1)(n-1)-sphere was contractible. (There is a big difference between “every \infty-category is an iterated colimit of copies of Δ 1\Delta^1 and Δ 0\Delta^0” (which is true) and “every \infty-category 𝒞\mathcal{C} is the colimit of the canonical diagram of shape (Δ n) /𝒞(\Delta_{\leq n})_{/\mathcal{C}}” which is what you would need to get a fully faithful embedding in presheaves on Δ n\Delta_{\leq n} - this is false for every finite nn.)

    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 \infty-category be a colimit of a diagram with objects [0],[1],[2][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 \infty-groupoid GG embeds as a constant presheaf, natural transformations KNGK \to NG correspond to morphisms colimKG\colim K \to G, and colim\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