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have added pointer to
Charles Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001), 973-1007 (arXiv:math/9811037, doi:10.1090/S0002-9947-00-02653-2)
Julia Bergner, Three models for the homotopy theory of homotopy theories, Topology Volume 46, Issue 4, September 2007, Pages 397-436 (arXiv:math/0504334, doi:10.1016/j.top.2007.03.002)
The nerve operation C↦n↦Core(C[n]) is a reflective embedding (∞,1)Cat→𝒫(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(Δ≤2)→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 Sp2→Δ2, Sp3→∂Δ3, and Δ0→J, 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:
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 ∂Δn→Δn, which would imply that the (n−1)-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 (Δ≤n)/𝒞” which is what you would need to get a fully faithful embedding in presheaves on Δ≤n - 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.
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 K→NG correspond to morphisms colimK→G, and colim converts a simplicial discrete space to the homotopy type of the corresponding simplicial set.
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