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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 15th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave 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](https://arxiv.org/abs/math/9811037), [doi:10.1090/S0002-9947-00-02653-2](https://doi.org/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](https://arxiv.org/abs/math/0504334), [doi:10.1016/j.top.2007.03.002](https://doi.org/10.1016/j.top.2007.03.002)) <a href="https://ncatlab.org/nlab/revision/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-categories/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/%28infinity%2C1%29-category+of+%28infinity%2C1%29-categories/11">v11</a>, <a href="https://ncatlab.org/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-categories">current</a>

   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)

   diff, v11, current

2. • CommentRowNumber2.
   • CommentAuthorHurkyl
   • CommentTimeFeb 22nd 2021
   • (edited Feb 22nd 2021)
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexThe 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?

   The nerve operation is a reflective embedding , where means the -category of -valued presheaves.

   I want to add the following claim here: the nerve factors through the reflective embedding ?

   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 , which I believe is true by expressing an -category as a colimit of categories, and rewrite categories as colimits of in a compatible way to make a single diagram. (using just and 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 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 , , and , where is the spine of and is the indiscrete simplicial space with 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 condition is redundant. Is that true?
   • Can we replace with the union of two 2-simplices expressing left and right inverses and ? Does this change the answer to the previous question?
3. • CommentRowNumber3.
   • CommentAuthorRuneHaugseng
   • CommentTimeFeb 23rd 2021
   • PermaLink
   Author: RuneHaugseng
   Format: MarkdownItexWhat 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.

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

   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.

4. • CommentRowNumber4.
   • CommentAuthorHurkyl
   • CommentTimeFeb 23rd 2021
   • (edited Feb 23rd 2021)
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexOh 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.

   Oh right, it’s not enough that an -category be a colimit of a diagram with objects , 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 embeds as a constant presheaf, natural transformations correspond to morphisms , and converts a simplicial discrete space to the homotopy type of the corresponding simplicial set.

5. • CommentRowNumber5.
   • CommentAuthorHurkyl
   • CommentTimeFeb 24th 2021
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexWhoops, forgot to alert I made the edit. <a href="https://ncatlab.org/nlab/revision/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-categories/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/%28infinity%2C1%29-category+of+%28infinity%2C1%29-categories/13">v13</a>, <a href="https://ncatlab.org/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-categories">current</a>

   Whoops, forgot to alert I made the edit.

   diff, v13, current