*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?

oughtto be presentable, regardless of whether it is. In recent work with Alex Chirvasitu, we convinced ourselves that (as we defined it) $\operatorname{Cat}_{c\otimes}(R)$ is cocomplete, but since we didn't need it, we didn't write up the proof. Note that you must assume that $R$ is commutative, or otherwise that it comes equipped with some extra structure; else $\operatorname{Mod}(R)$ and the rest are not monoidal categories (otherwise I'm not sure what it would mean to be enriched over $R$). $\endgroup$ – Theo Johnson-Freyd Jun 6 '11 at 1:30