Co/limits and 2-co/limits of categories in the $\infty$- and $(\infty,2)$-category of $\infty$-categories

Question

Recently, in a conversation with Gabriel, the following question came up:

Question. Do co/limits of categories taken in the $\infty$-category of $\infty$-categories agree with the usual co/limits taken in $\mathsf{Cats}$, the $1$-category of categories?

We then noticed there are a few variations on this question, due to all the higher-categorical stuff involved:

Question I. Let $\infty$-categories mean quasicategories as developed in Kerodon, and let $\mathsf{Cats}_\infty$ be the $\infty$-category of $\infty$-categories. Does the functor (I hope there's such a functor anyway)

$$\iota\colon\mathrm{N}_{\bullet}(\mathsf{Cats})\to\mathsf{Cats}_\infty$$ preserve/reflect $\infty$-categorical co/limits?

Question II. What analogous statements hold when working with $(\infty,2)$-categories, replacing $\mathrm{N}_{\bullet}(\mathsf{Cats})$ by the Duskin nerve of $\mathsf{Cats}$, replacing $\mathsf{Cats}_\infty$ by the $(\infty,2)$-category of $\infty$-categories, and working with the 10+ notions of $2$-co/limits, like "lax bilimit" or "pseudo limit", and the respective notions of $(\infty,2)$-co/limits?

Answer 1

The case of limits is very distinct from the case of colimits. Let me start with the case of colimits.

In general, none of these agree. There is a slight "refinement" of question I which makes it more plausible, namely replacing the $1$-category of categories with the $(2,1)$-category of categories, but it is still wrong.

Note that a negative answer to question I implies a negative answer to question II in full generality, as colimits can be realized as special cases of lax/pseudo/... colimits.

For the variant that I mentioned, the functor from categories to $\infty$-categories is fully faithful, and thus it does reflect the colimits that happen to remain $1$-categories (while this is not true if you ask about the $1$-category of categories).

Let me indicate one type of phenomenon that goes really wrong in all settings: constant colimits. Say you have a group $G$, and are looking at the constant diagram indexed by $BG$ at a $1$-category $C$. In the $1$-category of categories, its colmit is just $C$ (as is the case in any $1$-category!). The colimit in $Cat_\infty$ however, is $C\times BG$.

If $G$ is an ordinary group, this happens to still be a $1$-category and so is preserved under the variant that I mentioned. However, if $G$ was a more general, not -$1$-truncated $E_1$-group, then $BG$ can be an arbitrary space/$\infty$-groupoid, and so $C\times BG$ is no longer a $1$-category, and so the colimit simply cannot be preserved.

This is probably the simplest example, but even most pushouts tend to not be preserved - very special types of pushouts can be preserved, and more generally very special types of colimits can be preserved, but they should be thought of as the exception rather than the rule.

As I said, when it comes to $2$-categorical colimits, things get worse.

Now onto limits. Things are much better here. The way you formulated it, it's still generally not true: limits in the $1$-category of $1$-categories are way too strict. So the same "constant" example that I gave before works: in the $1$-category of categories, the limit of this example is just $C$, while in the $\infty$-category of $\infty$-categories, it becomes $C^{BG}$. You might now think that this also gives similar counterexamples in the proposed variant with funkier groups, but you'll now notice that for any $\infty$-category $D$, $Fun(D,C) = Fun(ho(D),C)$ and so the result is a $1$-category.

In fact, the inclusion from the $(2,1)$-category of categories inside $Cat_\infty$ admits a left adjoint ($ho(-)$) and thus preserves all limits. Furthermore, this turns out to be an $(\infty,2)$-categorical adjunction, and hence the inclusion actually preserves any type of $(\infty,2)$-categorical limit you might want to consider.