Example of a locally presentable $2$-category

Question

Background

Let $R$ be a (as always commutative) ring. Consider the $2$-category $\text{Cat}_{c\otimes}(R)$ of cocomplete $R$-linear tensor categories. There are various reasons why $\text{Cat}_{c\otimes}(R)$ can be seen as a categorified version of the category of $R$-algebras. For example, coproducts are categorified sums, tensor products are categorified products, and the imposed cocontinuity of the tensor product can be seen as a sort of distributive law. There is a $2$-functor $\text{Alg}(R) \to \text{Cat}_{c\otimes}(R), A \mapsto Mod(A)$ ("categorification"), which is $2$-left adjoint to $\text{Cat}_{c\otimes}(R) \to Alg(R), C \mapsto \text{End}(1_C)$ ("decategorification"). See also Lurie's article Tannaka duality for geometric stacks for evidence that that this category is important in algebraic geometry: The category of geometric stacks over $R$ embeds via quasi-coherent sheaves fully faithful into $\text{Cat}_{c\otimes}(R)$ (if we restrict to so called tame tensor functors). See also this entry in Todd Trimble's blog.

Question

Is $\text{Cat}_{c\otimes}(R)$ a locally presentable $2$-category?

Note that I don't want to consider it as a $(2,1)$-category here, but you may assume this if it is really necessary. To avoid set-theoretic difficulties in the following, perhaps we should restrict to categories with $\kappa$-small colimits for a fixed regular cardinal $\kappa$. Then $\text{Cat}_{c\otimes}(R)$ is probably $2$-cocomplete, see Mike Shulman's answer here. It already seems to be hard to describe colimits explicitly (see here).

There is a free cocomplete $R$-linear tensor category on one object, explicitly given by $\text{Mod}(R)^{\mathbb{N}}$ with a convolution tensor product. Imagine this as the categorified ring of power series over $R$. Thus $Hom(\text{Mod}(R)^{\mathbb{N}},-)$ is identified with the forgetful $2$-functor $\text{Cat}_{c\otimes}(R) \to \text{Cat}$, which is, however, not faithful since the data of a cocont. tensor functor does not only consist of a funcor, there is also the natural isomorphism expressing the compatibility with the tensor structure. Thus $\text{Mod}(R)^{\mathbb{N}}$ is not a generator. Remark that in the decategorified setting, the free $R$-algebra on one object, namely the polynomial algebra $R[x]$ is a generator of $\text{Alg}(R)$.

But maybe $\text{Cat}_{c\otimes}(R)$ is too big to be generated by a set of $\lambda$-presentable objects? If this is not the case, what is a reasonable full subcategory which contains the categories of quasi-coherent sheaves and is locally presentable as a $2$-category?

EDIT: Motivated by Jacob Lurie's answer, I would like to change the question to the following: Let $\lambda$ be a regular cardinal and $\text{Cat}_{c\otimes}^\lambda(R)$ denote the $2$-category of locally $\lambda$-presentable $R$-linear tensor categories (of course the definition of this includes that the $\lambda$-presentable objects form a submonoid with respect to $\otimes$). Is it a locally presentable $2$-category?

Answer 1

Your 2-category can be described as commutative algebras over "R-mod" in the setting of cocomplete categories (at least this is one meaning of the word "R-linear", which does not imply that the underlying category is abelian. If you want to stick to abelian categories the answer will be the same but require some more words.)

So a more basic question is: is the collection of cocomplete categories (maybe with some adjectives attached) locally presentable? If you fix a regular cardinal $\kappa$, require all categories to be generated by a set of $\kappa$-compact objects, and all functors to preserve $\kappa$-compact objects, then the answer is yes. For example, if $\kappa$ is uncountable, then this 2-category is equivalent (via the functor which formally adjoins $\kappa$-filtered colimits) to the 2-category of small categories which admit $\kappa$-small colimits, and functors which preserve $\kappa$-small colimits (this 2-category is "algebraic" in nature, albeit with respect to operations of (fixed) infinite arity).

If you don't restrict your functors to preserve $\kappa$-compact objects, then it's not reasonable for the underlying $(2,1)$-category to be locally presentable because it is not even locally small. Your post contains an example: there's a free $R$-linear tensor category on one generator, and the category of $R$-linear tensor functors from that to some target category is equivalent to the target category, which is typically not small. (There is probably some reasonable way to salvage the situation if you take advantage of the full 2-category structure: the morphism categories in your example will not be small but they are nevertheless accessible.)