2-colimits in the category of cocomplete categories

Question

Let us denote by $\text{Cat}_c$ the $2$-category, whose objects are cocomplete categories, morphisms are cocontinuous functors and morphisms are natural transformations. Is it then true that $\text{Cat}_c$ is $2$-cocomplete, i.e. has every $2$-functor $C : I \to \text{Cat}_c$, where $I$ is a small $1$-category, a $2$-colimit?

I have the following idea: Let $U :\text{Cat}_c \to \text{Cat}$ be the forgetful $2$-functor. Now $\text{Cat}$ is $2$-cocomplete; a comprehensive reference for this is "The stack of microlocal perverse sheaves" by Ingo Waschkies. Besides, $U$ has a $2$-left adjoint $\widehat{(- )} : \text{Cat} \to \text{Cat}_c$ (see here). So we may consider the cocomplete category $D:=\widehat{\text{colim} UC}$, but of course the functors $C_i \to D$ are not cocontinuous. Thus perhaps we have to define some reflective (and hence cocomplete) subcategory $D'$ of $D$, such that each $C_i \to D$ factors through a cocontinuous functor $C_i \to D'$. But I don't know how to define $D'$.

I'm also interested in colimits in similar $2$-categories, for example $\text{Cat}_{c\otimes}$, which consists of the cocomplete tensor categories. And actually I want something more, namely that these $2$-categories are $2$-locally presentable. Any hints and references are appreciated!

Answer 1

In order not to have to worry about size issues, I'm going to answer the following question instead:

For a (small) cardinal number $\kappa$, is the category of small categories with $\kappa$-small 2-colimits 2-cocomplete?

If you take $\kappa$ to be inaccessible, then this will correspond to your question, under a particular choice of foundations. I presume moreover that you mean "2-colimits" in the weak "up-to-equivalence" sense which the nLab uses (which 2-category theorists traditionally call "bicolimits").

The fact that the 2-category Cat of small categories is 2-cocomplete, in this sense, has been well-known to category theorists for decades. It is obvious that Cat is cocomplete as a 1-category (since it is locally finitely presentable), and since it is closed symmetric (cartesian) monoidal, it follows by general enriched category theory that it is cocomplete as a category enriched over itself. In the nLab terminology, it has all strict 2-colimits. We then observe that strict pseudo 2-limits, which are 2-limits that represent cones commuting up to isomorphism but satisfy their universal property up to isomorphism (rather than equivalence), are particular strict 2-limits. Since any strict pseudo 2-limit is also a (weak) 2-limit, Cat is 2-cocomplete.

Now as Zoran pointed out in the comments, there is a 2-monad on Cat whose algebras are categories with $\kappa$-small colimits; let us call this 2-monad $T$. The strict $T$-morphisms are functors which preserve colimits on-the-nose, while the pseudo $T$-morphisms are those which are $\kappa$-cocontinuous in the usual sense (preserve colimits up to isomorphism). Therefore, the question is whether the 2-category $T$-Alg of $T$-algebras and pseudo $T$-morphisms is 2-cocomplete.

The answer is yes: it was proven by Blackwell, Kelly, and Power in the paper "Two-dimensional monad theory" that for any 2-monad with a rank (preserving $\alpha$-filtered colimits for some $\alpha$) on a strictly 2-cocomplete (strict) 2-category, the 2-category $T$-Alg is (weakly) 2-cocomplete. The 2-monad $T$ has a rank (namely, $\kappa$, more or less), so their theorem applies. I believe this all works just as well in the enriched setting.