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Letting $\mathrm{Cat}$ denote the category of (small) categories and $\mathrm{Set}_\Delta$ the category of simplicial sets, it is well known that the nerve $N:\mathrm{Cat} \to \mathrm{Set}_\Delta$ admits a left adjoint which we can denote by $\mathrm{Ho}:\mathrm{Set}_\Delta \to \mathrm{Cat}$.

Endowing $\mathrm{Cat}$ with the unique (proper) model structure $[1]$ having as weak equivalences the usual equivalences of categories and $\mathrm{Set}_\Delta$ with the Joyal model structure one can readily check that $(\mathrm{Ho}, N)$ is a Quillen pair.

By work of Aaron Mazel-Gee $[2]$ this induces an adjunction $$i:\mathcal{C}\mathrm{at} \leftrightarrows\mathcal{C}\mathrm{at}_\infty:\mathrm{Ho},$$ between the $\infty$-categories underlying, respectively, the model categories $\mathrm{Cat}$ and $\mathrm{Set}_\Delta$ above. We can therefore conclude that $i: \mathcal{C}\mathrm{at} \to \mathcal{C}\mathrm{at}_\infty$ preserves limits (5.2.3.5 Higher Topos Theory).

However, I do not believe that $i$ preserves colimits. My argument goes as follows:

One can also present $\mathcal{C}\mathrm{at}_\infty$ as the $\infty$-category underlying the cartesian model structure on $\mathrm{Set}_\Delta^+$, the category of marked simplical sets. As this is combinatorial (3.1.3.7 HTT) and simplicial (3.1.4.4 HTT) $\mathcal{C}\mathrm{at}_\infty$ is presentable (A.3.7.6 HTT).

Similarly, in a note by Rezk $[3]$ it is shown that canonical model structure on $\mathrm{Cat}$ is cofibrantly generated and simplicial. I assume that $\mathrm{Cat}$ is locally presentable, though I do not know of a reference. Assuming that this is the case, $\mathcal{C}\mathrm{at}$ is also presentable.

By the adjoint functor theorem (5.5.2.9 HTT) $i$ preserves colimits if and only if $i$ admits a right adjoint. Let's denote such a candidate right adjoint by $R: \mathcal{C}\mathrm{at}_\infty \to \mathcal{C}\mathrm{at}$. Then in particular it must be that case that for each ordinary category $\mathrm{C}$ and each $\infty$-category $\mathcal{C}$, $$\mathrm{Map}( \mathrm{C}, \mathcal{C}) \simeq \mathrm{Map}(\mathrm{C}, R(\mathcal{C}).$$ Consider the case when $\mathrm{C} = [0]$ and $\mathcal{C}$ is an $\infty$-groupoid $X$. Then the above adjunction would require an equivalence of spaces $$X \simeq \iota R(X).$$ Since $R(X)$ is an ordinary category, $\iota R(X)$ is $1$-truncated and so such an equivalence cannot hold for general $X$.

What colimits are preserved by $i:\mathcal{C}\mathrm{at} \to \mathcal{C}\mathrm{at}_\infty$? In particular, which pushouts are preserved by $i$?

$[1]$: https://sbseminar.wordpress.com/2012/11/16/the-canonical-model-structure-on-cat/

$[2]$: Aaron Mazel-Gee, Quillen adjunctions induce adjunctions of quasicategories

$[3]$: Charles Rezk, A model category for categories

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    $\begingroup$ $\infty$-categories are generated by simplices under colimits, so $Cat$ cannot be closed under colimits $\endgroup$ – Denis Nardin Aug 15 '16 at 22:40
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    $\begingroup$ Welcome to MO Mark :-) (PS: did you have slides/notes for your recent talk at Leicester?) $\endgroup$ – David Roberts Aug 15 '16 at 23:26
  • $\begingroup$ @DenisNardin Thanks for the comment. I realised shortly after submitting that the answer to the question was in the negative, though for a different reason. I've edited the question to reflect this. Do you have a reference for the statement that $\mathcal{C}at_{oo}$ is generated by simplices? $\endgroup$ – Mark Penney Aug 15 '16 at 23:26
  • $\begingroup$ @DavidRoberts Thanks for the welcome David! I have slides that I would be happy to send you. $\endgroup$ – Mark Penney Aug 15 '16 at 23:30
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    $\begingroup$ Cat is locally presentable. Check out the examples in Adamek and Rosicky, right after the definition. $\endgroup$ – David White Aug 16 '16 at 1:15

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