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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 15th 2020

• CommentRowNumber2.
• CommentAuthorHurkyl
• CommentTime7 days ago
• (edited 7 days ago)

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

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

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 $[0], [1], [2]$, which I believe is true by expressing an $\infty$-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 $(\infty,1)Cat$ 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 $Sp^2 \to \Delta^2$, $Sp^3 \to \partial \Delta^3$, and $\Delta^0 \to J$, where $Sp^n$ is the spine of $\Delta^n$ and $J$ is the indiscrete simplicial space with $J_0$ 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 $Sp^3 \to \partial \Delta^3$ condition is redundant. Is that true?
• Can we replace $J$ with the union of two 2-simplices expressing left and right inverses $fg \simeq 1$ and $gh \simeq 1$? Does this change the answer to the previous question?
• CommentRowNumber3.
• CommentAuthorRuneHaugseng
• CommentTime7 days ago

What you’re suggesting is not correct: you have to consider the spine inclusions for all $n$ to describe $\infty$-categories. Otherwise, for instance, any $\infty$-groupoid would be local for $\partial \Delta^n \to \Delta^n$, which would imply that the $(n-1)$-sphere was contractible. (There is a big difference between “every $\infty$-category is an iterated colimit of copies of $\Delta^1$ and $\Delta^0$” (which is true) and “every $\infty$-category $\mathcal{C}$ is the colimit of the canonical diagram of shape $(\Delta_{\leq n})_{/\mathcal{C}}$” which is what you would need to get a fully faithful embedding in presheaves on $\Delta_{\leq n}$ - 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.

• CommentRowNumber4.
• CommentAuthorHurkyl
• CommentTime7 days ago
• (edited 7 days ago)

Oh right, it’s not enough that an $\infty$-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 $\infty$-groupoid $G$ embeds as a constant presheaf, natural transformations $K \to NG$ correspond to morphisms $\colim K \to G$, and $\colim$ converts a simplicial discrete space to the homotopy type of the corresponding simplicial set.

• CommentRowNumber5.
• CommentAuthorHurkyl
• CommentTime6 days ago