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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 $C \mapsto n \mapsto Core(C^{[n]})$ 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?
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 $n$ 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.

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 $\infty$-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.

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